arX

iv1

308

4008

v1 [

csA

I] 1

9 A

ug 2

013

A Literature Survey of Benchmark Functions For Global

Optimization Problems

Momin Jamillowastdagger Xin-She YangDagger lowastBlekinge Institute of TechnologySE-37179 Karlskrona Sweden

daggerHarman International Cooperate DivisionBecker-Goering Str 16 D-76307 Karlsbad Germany

E-mail mominjamilharmancomDaggerMiddlesex University

School of Science and TechnologyHendon Campus London NW4 4BT UKE-mail xin-sheyangmiddlesexacuk

Citation details

Momin Jamil and Xin-She Yang A literature survey of benchmark functions forglobal optimization problems Int Journal of Mathematical Modelling and

Numerical Optimisation Vol 4 No 2 pp 150ndash194 (2013)DOI 101504IJMMNO2013055204

Test functions are important to validate and compare the performance of optimizationalgorithms There have been many test or benchmark functions reported in the literaturehowever there is no standard list or set of benchmark functions Ideally test functionsshould have diverse properties so that can be truly useful to test new algorithms in anunbiased way For this purpose we have reviewed and compiled a rich set of 175 bench-mark functions for unconstrained optimization problems with diverse properties in termsof modality separability and valley landscape This is by far the most complete set offunctions so far in the literature and tt can be expected this complete set of functions canbe used for validation of new optimization in the future

1 Introduction

The test of reliability efficiency and validation of optimization algorithms is frequentlycarried out by using a chosen set of common standard benchmarks or test functions fromthe literature The number of test functions in most papers varied from a few to abouttwo dozens Ideally the test functions used should be diverse and unbiased however thereis no agreed set of test functions in the literature Therefore the major aim of this paperis to review and compile the most complete set of test functions that we can find from allthe available literature so that they can be used for future validation and comparison ofoptimization algorithms

For any new optimization it is essential to validate its performance and compare withother existing algorithms over a good set of test functions A common practice followedby many researches is to compare different algorithms on a large test set especially when

1

the test involves function optimization (Gordon 1993 Whitley 1996) However it must benoted that effectiveness of one algorithm against others simply cannot be measured by theproblems that it solves if the the set of problems are too specialized and without diverseproperties Therefore in order to evaluate an algorithm one must identify the kind ofproblems where it performs better compared to others This helps in characterizing thetype of problems for which an algorithm is suitable This is only possible if the test suite islarge enough to include a wide variety of problems such as unimodal multimodal regularirregular separable non-separable and multi-dimensional problems

Many test functions may be scattered in different textbooks in individual researcharticles or at different web sites Therefore searching for a single source of test functionwith a wide variety of characteristics is a cumbersome and tedious task The most notableattempts to assemble global optimization test problems can be found in [4 7 8 15 20 2928 32 33 52 64 65 66 74 77 78 82 83 84 86] Online collections of test problems alsoexist such as the GLOBAL library at the cross-entropy toolbox [18] GAMS World [36]CUTE [41] global optimization test problems collection by Hedar [43] collection of testfunctions [5 37 48 53 54 55 56 57 58 59] a collection of continuous global optimizationtest problems COCONUT [61] and a subset of commonly used test functions [88] Thismotivates us to carry out a thorough analysis and compile a comprehensive collection ofunconstrained optimization test problems

In general unconstrained problems can be classified into two categories test functionsand real-world problems Test functions are artificial problems and can be used to evaluatethe behavior of an algorithm in sometimes diverse and difficult situations Artificial prob-lems may include single global minimum single or multiple global minima in the presence ofmany local minima long narrow valleys null-space effects and flat surfaces These problemscan be easily manipulated and modified to test the algorithms in diverse scenarios On theother hand real-world problems originate from different fields such as physics chemistryengineering mathematics etc These problems are hard to manipulate and may containcomplicated algebraic or differential expressions and may require a significant amount ofdata to compile A collection of real-world unstrained optimization problems can be foundin [7 8]

In this present work we will focus on the test function benchmarks and their diverseproperties such as modality and separability A function with more than one local optimumis called multimodal These functions are used to test the ability of an algorithm to escapefrom any local minimum If the exploration process of an algorithm is poorly designedthen it cannot search the function landscape effectively This in turn leads to an algorithmgetting stuck at a local minimum Multi-modal functions with many local minima are amongthe most difficult class of problems for many algorithms Functions with flat surfaces pose adifficulty for the algorithms since the flatness of the function does not give the algorithm anyinformation to direct the search process towards the minima (Stepint Matyas PowerSum)Another group of test problems is formulated by separable and non-separable functionsAccording to [16] the dimensionality of the search space is an important issue with theproblem In some functions the area that contains that global minima are very small whencompared to the whole search space such as Easom Michalewicz (m=10) and Powell Forproblems such as Perm Kowalik and Schaffer the global minimum is located very close tothe local minima If the algorithm cannot keep up the direction changes in the functionswith a narrow curved valley in case of functions like Beale Colville or cannot explore thesearch space effectively in case of function like Pen Holder Testtube-Holder having multipleglobal minima the algoritm will fail for these kinds of problems Another problem that

2

algorithms may suffer is the scaling problem with many orders of magnitude differencesbetween the domain and the function hyper-surface [47] such as Goldstein-Price and Trid

2 Characteristics of Test Functions

The goal of any global optimization (GO) is to find the best possible solutions xlowast from aset X according to a set of criteria F = f1 f2 middot middot middot fn These criteria are called objectivefunctions expressed in the form of mathematical functions An objective function is amathematical function f D sub realn rarr real subject to additional constraints The set Dis referred to as the set of feasible points in a search space In the case of optimizing asingle criterion f an optimum is either its maximum or minimum The global optimizationproblems are often defined as minimization problems however these problems can be easilyconverted to maximization problems by negating f A general global optimum problem canbe defined as follows

minimizex

f(x) (1)

The true optimal solution of an optimization problem may be a set of xlowast isin D of all optimalpoints in D rather than a single minimum or maximum value in some cases There couldbe multiple even an infinite number of optimal solutions depending on the domain ofthe search space The tasks of any good global optimization algorithm is to find globallyoptimal or at least sub-optimal solutions The objective functions could be characterized ascontinuous discontinuous linear non-linear convex non-conxex unimodal multimodalseparable1 and non-separable

According to [20] it is important to ask the following two questions before start solvingan optimization problem (i) What aspects of the function landscape make the optimiza-tion process difficult (ii) What type of a priori knowledge is most effective for searchingparticular types of function landscape In order to answer these questions benchmarkfunctions can be classified in terms of features like modality basins valleys separabilityand dimensionality [87]

21 Modality

The number of ambiguous peaks in the function landscape corresponds to the modality of afunction If algorithms encounters these peaks during a search process there is a tendencythat the algorithm may be trapped in one of such peaks This will have a negative impacton the search process as this can direct the search away from the true optimal solutions

22 Basins

A relatively steep decline surrounding a large area is called a basin Optimization algorithmscan be easily attracted to such regions Once in these regions the search process of analgorithm is severely hampered This is due to lack of information to direct the searchprocess towards the minimum According to [20] a basin corresponds to the plateau for amaximization problem and a problem can have multiple plateaus

1In this paper partially separable functions are also considered as separable function

3

23 Valleys

A valley occurs when a narrow area of little change is surrounded by regions of steep descent[20] As with the basins minimizers are initially attracted to this region The progress of asearch process of an algorithm may be slowed down considerably on the floor of the valley

24 Separability

The separability is a measure of difficulty of different benchmark functions In generalseparable functions are relatively easy to solve when compared with their inseperable coun-terpart because each variable of a function is independent of the other variables If all theparameters or variables are independent then a sequence of n independent optimizationprocesses can be performed As a result each design variable or parameter can be opti-mized independently According to [74] the general condition of separability to see if thefunction is easy to optimize or not is given as

partf(x)

partxi= g(xi)h(x) (2)

where g(xi) means any function of xi only and h(x) any function of any x If this conditionis satisfied the function is called partially separable and easy to optimize because solutionsfor each xi can be obtained independently of all the other parameters This separabilitycondition can be illustrated by the following two examples

For example function (f105) is not separable because it does not satisfy the condition(2)

partf105(x1 x2)

partx1= 400(x21 minus x2)x1 minus 2x1 minus 2

partf105(x1 x2)

partx2= minus200(x21 minus x2)

On the other hand the sphere function (f137) with two variables can indeed satisfy theabove condition (2) as shown below

partf137(x1 x2)

partx1= 2x1

partf137(x1 x2)

partx2= 2x2

where h(x) is regarded as 1In [16] the formal definition of separability is given as

arg minimizex1xp

f(x1 xp) =(

arg minimizex1

f(x1 )

arg minimizexp

f( xp))

(3)

In other words a function of p variables is called separable if it can written as a sumof p functions of just one variable [16] On the other hand a function is called non-separable if its variables show inter-relation among themselves or are not independent Ifthe objective function variables are independent of each other then the objective functionscan be decomposed into sub-objective functions Then each of these sub-objectives involvesonly one decision variable while treating all the others as constant and can be expressed as

4

f(x1 x2 middot middot middot xp) =psum

i=1

fi(xi) (4)

25 Dimensionality

The difficulty of a problem generally increases with its dimensionality According to [8790] as the number of parameters or dimension increases the search space also increasesexponentially For highly nonlinear problems this dimensionality may be a significantbarrier for almost all optimization algorithms

3 Benchmark Test Functions for Global Optimization

Now we present a collection of 175 unconstrained optimization test problems which canbe used to validate the performance of optimization algorithms The dimensions problemdomain size and optimal solution are denoted by D Lb le xi le Ub and f(xlowast) = f(x1 xn)respectively The symbols Lb and Ub represent lower upper bound of the variables re-spectively It is worth noting that in several cases the optimal solution vectors and theircorresponding solutions are known only as numerical approximations

1 Ackley 1 Function [9](Continuous Differentiable Non-separable Scalable Multi-modal)

f1(x) = minus20eminus002

radic

Dminus1sumD

i=1 x2i minus eD

minus1sumD

i=1 cos(2πxi) + 20 + e

subject to minus35 le xi le 35 The global minima is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

2 Ackley 2 Function [1] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f2(x) = minus200eminus002radic

x21+x2

2

subject to minus32 le xi le 32 The global minimum is located at origin xlowast = (0 0)f(xlowast) = minus200

3 Ackley 3 Function [1] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f3(x) = 200eminus002radic

x21+x2

2 + 5ecos(3x1)+sin(3x2)

subject to minus32 le xi le 32 The global minimum is located at xlowast = (0asymp minus04)f(xlowast) asymp minus2191418

5

4 Ackley 4 or Modified Ackley Function (Continuous Differentiable Non-SeparableScalable Multimodal)

f4(x) =Dsum

i=1

(

eminus02radic

x2i + x2i+1 + 3 (cos(2xi) + sin(2xi+1))

)

subject to minus35 le xi le 35 It is highly multimodal function with two global minimumclose to origin

x = f(minus1479252minus0739807 1479252minus0739807) f(xlowast) = minus3917275

5 Adjiman Function [2](Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f5(x) = cos(x1)sin(x2)minusx1

(x22 + 1)

subject to minus1 le x1 le 2 minus1 le x2 le 1 The global minimum is located at xlowast =(2 010578) f(xlowast) = minus202181

6 Alpine 1 Function [69](Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f6(x) =

Dsum

i=1

∣

∣

∣xisin(xi) + 01xi

∣

∣

∣

subject to minus10 le xi le 10 The global minimum is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

7 Alpine 2 Function [21] (Continuous Differentiable Separable Scalable Multi-modal)

f7(x) =

Dprod

i=1

radicxisin(xi)

subject to 0 le xi le 10 The global minimum is located at xlowast = (7917 middot middot middot 7917)f(xlowast) = 2808D

8 Brad Function [17] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f8(x) =

15sum

i=1

[

yi minus x1 minus uivix2 +wix3

]2

where ui = i vi = 16 minus i wi = min(ui vi) and y = yi = [014 018 022 025 029

032 035 039 037 058 073 096 134 210 439]T It is subject to minus025 le x1 le025 001 le x2 x3 le 25 The global minimum is located at xlowast = (00824 1133 23437)f(xlowast) = 000821487

6

9 Bartels Conn Function (Continuous Non-differentiable Non-Separable Non-ScalableMultimodal)

f9(x) =∣

∣x21 + x22 + x1x2∣

∣+∣

∣sin(x1)∣

∣+∣

∣cos(x2)∣

∣

subject to minus500 le xi le 500 The global minimum is located at xlowast = (0 0) f(xlowast) = 1

10 Beale Function (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f10(x) = (15minus x1 + x1x2)2 + (225 minus x1 + x1x

22)

2

+(2625 minus x1 + x1x32)

2

subject to minus45 le xi le 45 The global minimum is located at xlowast = (3 05) f(xlowast) = 0

11 Biggs EXP2 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f11(x) =

10sum

i=1

(

eminustix1 minus 5eminustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10) f(xlowast) = 0

12 Biggs EXP3 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f12(x) =

10sum

i=1

(

eminustix1 minus x3eminustix2 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 5) f(xlowast) = 0

13 Biggs EXP4 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f13(x) =

10sum

i=1

(

x3eminustix1 minus x4e

minustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 1 5) f(xlowast) = 0

14 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f14(x) =

11sum

i=1

(

x3eminustix1 minus x4e

minustix2 + 3eminustix5 minus yi)2

7

where ti = 01i yi = eminusti minus 5e10ti + 3eminus4ti It is subject to 0 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4) f(xlowast) = 0

15 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f15(x) =

13sum

i=1

(

x3eminustix1 minus x4e

minustix2 + x6eminustix5 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti +3eminus4ti It is subject to minus20 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4 3) f(xlowast) = 0

16 Bird Function [58] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f16(x) = sin(x1)e(1minuscos(x2))2 + cos(x2)e

(1minussin(x1))2 + (x1 minus x2)2

subject to minus2π le xi le 2π The global minimum is located at xlowast = (470104315294)(minus158214 minus313024) f(xlowast) = minus106764537

17 Bohachevsky 1 Function [14] (Continuous Differentiable Separable Non-ScalableMultimodal)

f17(x) = x21 + 2x22 minus 03cos(3πx1)

minus04cos(4πx2) + 07

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

18 Bohachevsky 2 Function [14] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f18(x) = x21 + 2x22 minus 03cos(3πx1) middot 04cos(4πx2)+03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

19 Bohachevsky 3 Function [14] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f19(x) = x21 + 2x22 minus 03cos(3πx1 + 4πx2) + 03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

8

20 Booth Function (Continuous Differentiable Non-separable Non-Scalable Uni-modal)

f20(x) = (x1 + 2x2 minus 7)2 + (2x1 + x2 minus 5)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(1 3) f(xlowast) = 0

21 Box-Betts Quadratic Sum Function [4] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f21(x) =Dminus1sum

i=0

g(xi)2

where

g(x) = eminus01(i+1)x1 minus eminus01(i+1)x2 minus e[(minus01(i+1))minuseminus(i+1) ]x3

subject to 09 le x1 le 12 9 le x2 le 112 09 le x2 le 12 The global minimum islocated at xlowast = f(1 10 1) f(xlowast) = 0

22 Branin RCOS Function [15] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f22(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) + 10

with domain minus5 le x1 le 10 0 le x1 le 15 It has three global minima at xlowast =f(minusπ 12275 π 2275 3π 2425) f(xlowast) = 03978873

23 Branin RCOS 2 Function [60] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f23(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) cos(x2) ln(x21 + x22 + 1) + 10

with domain minus5 le xi le 15 The global minimum is located at xlowast = f(minus32 1253)f(xlowast) = 5559037

24 Brent Function [15] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f24(x) = (x1 + 10)2 + (x2 + 10)2 + eminusx21minusx2

2 (2)

with domain minus10 le xi le 10 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

9

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

algorithms may suffer is the scaling problem with many orders of magnitude differencesbetween the domain and the function hyper-surface [47] such as Goldstein-Price and Trid

2 Characteristics of Test Functions

The goal of any global optimization (GO) is to find the best possible solutions xlowast from aset X according to a set of criteria F = f1 f2 middot middot middot fn These criteria are called objectivefunctions expressed in the form of mathematical functions An objective function is amathematical function f D sub realn rarr real subject to additional constraints The set Dis referred to as the set of feasible points in a search space In the case of optimizing asingle criterion f an optimum is either its maximum or minimum The global optimizationproblems are often defined as minimization problems however these problems can be easilyconverted to maximization problems by negating f A general global optimum problem canbe defined as follows

minimizex

f(x) (1)

The true optimal solution of an optimization problem may be a set of xlowast isin D of all optimalpoints in D rather than a single minimum or maximum value in some cases There couldbe multiple even an infinite number of optimal solutions depending on the domain ofthe search space The tasks of any good global optimization algorithm is to find globallyoptimal or at least sub-optimal solutions The objective functions could be characterized ascontinuous discontinuous linear non-linear convex non-conxex unimodal multimodalseparable1 and non-separable

According to [20] it is important to ask the following two questions before start solvingan optimization problem (i) What aspects of the function landscape make the optimiza-tion process difficult (ii) What type of a priori knowledge is most effective for searchingparticular types of function landscape In order to answer these questions benchmarkfunctions can be classified in terms of features like modality basins valleys separabilityand dimensionality [87]

21 Modality

The number of ambiguous peaks in the function landscape corresponds to the modality of afunction If algorithms encounters these peaks during a search process there is a tendencythat the algorithm may be trapped in one of such peaks This will have a negative impacton the search process as this can direct the search away from the true optimal solutions

22 Basins

A relatively steep decline surrounding a large area is called a basin Optimization algorithmscan be easily attracted to such regions Once in these regions the search process of analgorithm is severely hampered This is due to lack of information to direct the searchprocess towards the minimum According to [20] a basin corresponds to the plateau for amaximization problem and a problem can have multiple plateaus

1In this paper partially separable functions are also considered as separable function

3

23 Valleys

A valley occurs when a narrow area of little change is surrounded by regions of steep descent[20] As with the basins minimizers are initially attracted to this region The progress of asearch process of an algorithm may be slowed down considerably on the floor of the valley

24 Separability

The separability is a measure of difficulty of different benchmark functions In generalseparable functions are relatively easy to solve when compared with their inseperable coun-terpart because each variable of a function is independent of the other variables If all theparameters or variables are independent then a sequence of n independent optimizationprocesses can be performed As a result each design variable or parameter can be opti-mized independently According to [74] the general condition of separability to see if thefunction is easy to optimize or not is given as

partf(x)

partxi= g(xi)h(x) (2)

where g(xi) means any function of xi only and h(x) any function of any x If this conditionis satisfied the function is called partially separable and easy to optimize because solutionsfor each xi can be obtained independently of all the other parameters This separabilitycondition can be illustrated by the following two examples

For example function (f105) is not separable because it does not satisfy the condition(2)

partf105(x1 x2)

partx1= 400(x21 minus x2)x1 minus 2x1 minus 2

partf105(x1 x2)

partx2= minus200(x21 minus x2)

On the other hand the sphere function (f137) with two variables can indeed satisfy theabove condition (2) as shown below

partf137(x1 x2)

partx1= 2x1

partf137(x1 x2)

partx2= 2x2

where h(x) is regarded as 1In [16] the formal definition of separability is given as

arg minimizex1xp

f(x1 xp) =(

arg minimizex1

f(x1 )

arg minimizexp

f( xp))

(3)

In other words a function of p variables is called separable if it can written as a sumof p functions of just one variable [16] On the other hand a function is called non-separable if its variables show inter-relation among themselves or are not independent Ifthe objective function variables are independent of each other then the objective functionscan be decomposed into sub-objective functions Then each of these sub-objectives involvesonly one decision variable while treating all the others as constant and can be expressed as

4

f(x1 x2 middot middot middot xp) =psum

i=1

fi(xi) (4)

25 Dimensionality

The difficulty of a problem generally increases with its dimensionality According to [8790] as the number of parameters or dimension increases the search space also increasesexponentially For highly nonlinear problems this dimensionality may be a significantbarrier for almost all optimization algorithms

3 Benchmark Test Functions for Global Optimization

Now we present a collection of 175 unconstrained optimization test problems which canbe used to validate the performance of optimization algorithms The dimensions problemdomain size and optimal solution are denoted by D Lb le xi le Ub and f(xlowast) = f(x1 xn)respectively The symbols Lb and Ub represent lower upper bound of the variables re-spectively It is worth noting that in several cases the optimal solution vectors and theircorresponding solutions are known only as numerical approximations

1 Ackley 1 Function [9](Continuous Differentiable Non-separable Scalable Multi-modal)

f1(x) = minus20eminus002

radic

Dminus1sumD

i=1 x2i minus eD

minus1sumD

i=1 cos(2πxi) + 20 + e

subject to minus35 le xi le 35 The global minima is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

2 Ackley 2 Function [1] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f2(x) = minus200eminus002radic

x21+x2

2

subject to minus32 le xi le 32 The global minimum is located at origin xlowast = (0 0)f(xlowast) = minus200

3 Ackley 3 Function [1] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f3(x) = 200eminus002radic

x21+x2

2 + 5ecos(3x1)+sin(3x2)

subject to minus32 le xi le 32 The global minimum is located at xlowast = (0asymp minus04)f(xlowast) asymp minus2191418

5

4 Ackley 4 or Modified Ackley Function (Continuous Differentiable Non-SeparableScalable Multimodal)

f4(x) =Dsum

i=1

(

eminus02radic

x2i + x2i+1 + 3 (cos(2xi) + sin(2xi+1))

)

subject to minus35 le xi le 35 It is highly multimodal function with two global minimumclose to origin

x = f(minus1479252minus0739807 1479252minus0739807) f(xlowast) = minus3917275

5 Adjiman Function [2](Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f5(x) = cos(x1)sin(x2)minusx1

(x22 + 1)

subject to minus1 le x1 le 2 minus1 le x2 le 1 The global minimum is located at xlowast =(2 010578) f(xlowast) = minus202181

6 Alpine 1 Function [69](Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f6(x) =

Dsum

i=1

∣

∣

∣xisin(xi) + 01xi

∣

∣

∣

subject to minus10 le xi le 10 The global minimum is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

7 Alpine 2 Function [21] (Continuous Differentiable Separable Scalable Multi-modal)

f7(x) =

Dprod

i=1

radicxisin(xi)

subject to 0 le xi le 10 The global minimum is located at xlowast = (7917 middot middot middot 7917)f(xlowast) = 2808D

8 Brad Function [17] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f8(x) =

15sum

i=1

[

yi minus x1 minus uivix2 +wix3

]2

where ui = i vi = 16 minus i wi = min(ui vi) and y = yi = [014 018 022 025 029

032 035 039 037 058 073 096 134 210 439]T It is subject to minus025 le x1 le025 001 le x2 x3 le 25 The global minimum is located at xlowast = (00824 1133 23437)f(xlowast) = 000821487

6

9 Bartels Conn Function (Continuous Non-differentiable Non-Separable Non-ScalableMultimodal)

f9(x) =∣

∣x21 + x22 + x1x2∣

∣+∣

∣sin(x1)∣

∣+∣

∣cos(x2)∣

∣

subject to minus500 le xi le 500 The global minimum is located at xlowast = (0 0) f(xlowast) = 1

10 Beale Function (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f10(x) = (15minus x1 + x1x2)2 + (225 minus x1 + x1x

22)

2

+(2625 minus x1 + x1x32)

2

subject to minus45 le xi le 45 The global minimum is located at xlowast = (3 05) f(xlowast) = 0

11 Biggs EXP2 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f11(x) =

10sum

i=1

(

eminustix1 minus 5eminustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10) f(xlowast) = 0

12 Biggs EXP3 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f12(x) =

10sum

i=1

(

eminustix1 minus x3eminustix2 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 5) f(xlowast) = 0

13 Biggs EXP4 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f13(x) =

10sum

i=1

(

x3eminustix1 minus x4e

minustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 1 5) f(xlowast) = 0

14 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f14(x) =

11sum

i=1

(

x3eminustix1 minus x4e

minustix2 + 3eminustix5 minus yi)2

7

where ti = 01i yi = eminusti minus 5e10ti + 3eminus4ti It is subject to 0 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4) f(xlowast) = 0

15 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f15(x) =

13sum

i=1

(

x3eminustix1 minus x4e

minustix2 + x6eminustix5 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti +3eminus4ti It is subject to minus20 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4 3) f(xlowast) = 0

16 Bird Function [58] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f16(x) = sin(x1)e(1minuscos(x2))2 + cos(x2)e

(1minussin(x1))2 + (x1 minus x2)2

subject to minus2π le xi le 2π The global minimum is located at xlowast = (470104315294)(minus158214 minus313024) f(xlowast) = minus106764537

17 Bohachevsky 1 Function [14] (Continuous Differentiable Separable Non-ScalableMultimodal)

f17(x) = x21 + 2x22 minus 03cos(3πx1)

minus04cos(4πx2) + 07

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

18 Bohachevsky 2 Function [14] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f18(x) = x21 + 2x22 minus 03cos(3πx1) middot 04cos(4πx2)+03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

19 Bohachevsky 3 Function [14] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f19(x) = x21 + 2x22 minus 03cos(3πx1 + 4πx2) + 03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

8

20 Booth Function (Continuous Differentiable Non-separable Non-Scalable Uni-modal)

f20(x) = (x1 + 2x2 minus 7)2 + (2x1 + x2 minus 5)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(1 3) f(xlowast) = 0

21 Box-Betts Quadratic Sum Function [4] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f21(x) =Dminus1sum

i=0

g(xi)2

where

g(x) = eminus01(i+1)x1 minus eminus01(i+1)x2 minus e[(minus01(i+1))minuseminus(i+1) ]x3

subject to 09 le x1 le 12 9 le x2 le 112 09 le x2 le 12 The global minimum islocated at xlowast = f(1 10 1) f(xlowast) = 0

22 Branin RCOS Function [15] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f22(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) + 10

with domain minus5 le x1 le 10 0 le x1 le 15 It has three global minima at xlowast =f(minusπ 12275 π 2275 3π 2425) f(xlowast) = 03978873

23 Branin RCOS 2 Function [60] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f23(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) cos(x2) ln(x21 + x22 + 1) + 10

with domain minus5 le xi le 15 The global minimum is located at xlowast = f(minus32 1253)f(xlowast) = 5559037

24 Brent Function [15] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f24(x) = (x1 + 10)2 + (x2 + 10)2 + eminusx21minusx2

2 (2)

with domain minus10 le xi le 10 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

9

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

23 Valleys

A valley occurs when a narrow area of little change is surrounded by regions of steep descent[20] As with the basins minimizers are initially attracted to this region The progress of asearch process of an algorithm may be slowed down considerably on the floor of the valley

24 Separability

The separability is a measure of difficulty of different benchmark functions In generalseparable functions are relatively easy to solve when compared with their inseperable coun-terpart because each variable of a function is independent of the other variables If all theparameters or variables are independent then a sequence of n independent optimizationprocesses can be performed As a result each design variable or parameter can be opti-mized independently According to [74] the general condition of separability to see if thefunction is easy to optimize or not is given as

partf(x)

partxi= g(xi)h(x) (2)

where g(xi) means any function of xi only and h(x) any function of any x If this conditionis satisfied the function is called partially separable and easy to optimize because solutionsfor each xi can be obtained independently of all the other parameters This separabilitycondition can be illustrated by the following two examples

For example function (f105) is not separable because it does not satisfy the condition(2)

partf105(x1 x2)

partx1= 400(x21 minus x2)x1 minus 2x1 minus 2

partf105(x1 x2)

partx2= minus200(x21 minus x2)

On the other hand the sphere function (f137) with two variables can indeed satisfy theabove condition (2) as shown below

partf137(x1 x2)

partx1= 2x1

partf137(x1 x2)

partx2= 2x2

where h(x) is regarded as 1In [16] the formal definition of separability is given as

arg minimizex1xp

f(x1 xp) =(

arg minimizex1

f(x1 )

arg minimizexp

f( xp))

(3)

In other words a function of p variables is called separable if it can written as a sumof p functions of just one variable [16] On the other hand a function is called non-separable if its variables show inter-relation among themselves or are not independent Ifthe objective function variables are independent of each other then the objective functionscan be decomposed into sub-objective functions Then each of these sub-objectives involvesonly one decision variable while treating all the others as constant and can be expressed as

4

f(x1 x2 middot middot middot xp) =psum

i=1

fi(xi) (4)

25 Dimensionality

The difficulty of a problem generally increases with its dimensionality According to [8790] as the number of parameters or dimension increases the search space also increasesexponentially For highly nonlinear problems this dimensionality may be a significantbarrier for almost all optimization algorithms

3 Benchmark Test Functions for Global Optimization

Now we present a collection of 175 unconstrained optimization test problems which canbe used to validate the performance of optimization algorithms The dimensions problemdomain size and optimal solution are denoted by D Lb le xi le Ub and f(xlowast) = f(x1 xn)respectively The symbols Lb and Ub represent lower upper bound of the variables re-spectively It is worth noting that in several cases the optimal solution vectors and theircorresponding solutions are known only as numerical approximations

1 Ackley 1 Function [9](Continuous Differentiable Non-separable Scalable Multi-modal)

f1(x) = minus20eminus002

radic

Dminus1sumD

i=1 x2i minus eD

minus1sumD

i=1 cos(2πxi) + 20 + e

subject to minus35 le xi le 35 The global minima is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

2 Ackley 2 Function [1] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f2(x) = minus200eminus002radic

x21+x2

2

subject to minus32 le xi le 32 The global minimum is located at origin xlowast = (0 0)f(xlowast) = minus200

3 Ackley 3 Function [1] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f3(x) = 200eminus002radic

x21+x2

2 + 5ecos(3x1)+sin(3x2)

subject to minus32 le xi le 32 The global minimum is located at xlowast = (0asymp minus04)f(xlowast) asymp minus2191418

5

4 Ackley 4 or Modified Ackley Function (Continuous Differentiable Non-SeparableScalable Multimodal)

f4(x) =Dsum

i=1

(

eminus02radic

x2i + x2i+1 + 3 (cos(2xi) + sin(2xi+1))

)

subject to minus35 le xi le 35 It is highly multimodal function with two global minimumclose to origin

x = f(minus1479252minus0739807 1479252minus0739807) f(xlowast) = minus3917275

5 Adjiman Function [2](Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f5(x) = cos(x1)sin(x2)minusx1

(x22 + 1)

subject to minus1 le x1 le 2 minus1 le x2 le 1 The global minimum is located at xlowast =(2 010578) f(xlowast) = minus202181

6 Alpine 1 Function [69](Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f6(x) =

Dsum

i=1

∣

∣

∣xisin(xi) + 01xi

∣

∣

∣

subject to minus10 le xi le 10 The global minimum is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

7 Alpine 2 Function [21] (Continuous Differentiable Separable Scalable Multi-modal)

f7(x) =

Dprod

i=1

radicxisin(xi)

subject to 0 le xi le 10 The global minimum is located at xlowast = (7917 middot middot middot 7917)f(xlowast) = 2808D

8 Brad Function [17] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f8(x) =

15sum

i=1

[

yi minus x1 minus uivix2 +wix3

]2

where ui = i vi = 16 minus i wi = min(ui vi) and y = yi = [014 018 022 025 029

032 035 039 037 058 073 096 134 210 439]T It is subject to minus025 le x1 le025 001 le x2 x3 le 25 The global minimum is located at xlowast = (00824 1133 23437)f(xlowast) = 000821487

6

9 Bartels Conn Function (Continuous Non-differentiable Non-Separable Non-ScalableMultimodal)

f9(x) =∣

∣x21 + x22 + x1x2∣

∣+∣

∣sin(x1)∣

∣+∣

∣cos(x2)∣

∣

subject to minus500 le xi le 500 The global minimum is located at xlowast = (0 0) f(xlowast) = 1

10 Beale Function (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f10(x) = (15minus x1 + x1x2)2 + (225 minus x1 + x1x

22)

2

+(2625 minus x1 + x1x32)

2

subject to minus45 le xi le 45 The global minimum is located at xlowast = (3 05) f(xlowast) = 0

11 Biggs EXP2 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f11(x) =

10sum

i=1

(

eminustix1 minus 5eminustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10) f(xlowast) = 0

12 Biggs EXP3 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f12(x) =

10sum

i=1

(

eminustix1 minus x3eminustix2 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 5) f(xlowast) = 0

13 Biggs EXP4 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f13(x) =

10sum

i=1

(

x3eminustix1 minus x4e

minustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 1 5) f(xlowast) = 0

14 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f14(x) =

11sum

i=1

(

x3eminustix1 minus x4e

minustix2 + 3eminustix5 minus yi)2

7

where ti = 01i yi = eminusti minus 5e10ti + 3eminus4ti It is subject to 0 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4) f(xlowast) = 0

15 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f15(x) =

13sum

i=1

(

x3eminustix1 minus x4e

minustix2 + x6eminustix5 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti +3eminus4ti It is subject to minus20 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4 3) f(xlowast) = 0

16 Bird Function [58] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f16(x) = sin(x1)e(1minuscos(x2))2 + cos(x2)e

(1minussin(x1))2 + (x1 minus x2)2

subject to minus2π le xi le 2π The global minimum is located at xlowast = (470104315294)(minus158214 minus313024) f(xlowast) = minus106764537

17 Bohachevsky 1 Function [14] (Continuous Differentiable Separable Non-ScalableMultimodal)

f17(x) = x21 + 2x22 minus 03cos(3πx1)

minus04cos(4πx2) + 07

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

18 Bohachevsky 2 Function [14] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f18(x) = x21 + 2x22 minus 03cos(3πx1) middot 04cos(4πx2)+03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

19 Bohachevsky 3 Function [14] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f19(x) = x21 + 2x22 minus 03cos(3πx1 + 4πx2) + 03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

8

20 Booth Function (Continuous Differentiable Non-separable Non-Scalable Uni-modal)

f20(x) = (x1 + 2x2 minus 7)2 + (2x1 + x2 minus 5)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(1 3) f(xlowast) = 0

21 Box-Betts Quadratic Sum Function [4] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f21(x) =Dminus1sum

i=0

g(xi)2

where

g(x) = eminus01(i+1)x1 minus eminus01(i+1)x2 minus e[(minus01(i+1))minuseminus(i+1) ]x3

subject to 09 le x1 le 12 9 le x2 le 112 09 le x2 le 12 The global minimum islocated at xlowast = f(1 10 1) f(xlowast) = 0

22 Branin RCOS Function [15] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f22(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) + 10

with domain minus5 le x1 le 10 0 le x1 le 15 It has three global minima at xlowast =f(minusπ 12275 π 2275 3π 2425) f(xlowast) = 03978873

23 Branin RCOS 2 Function [60] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f23(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) cos(x2) ln(x21 + x22 + 1) + 10

with domain minus5 le xi le 15 The global minimum is located at xlowast = f(minus32 1253)f(xlowast) = 5559037

24 Brent Function [15] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f24(x) = (x1 + 10)2 + (x2 + 10)2 + eminusx21minusx2

2 (2)

with domain minus10 le xi le 10 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

9

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

f(x1 x2 middot middot middot xp) =psum

i=1

fi(xi) (4)

25 Dimensionality

The difficulty of a problem generally increases with its dimensionality According to [8790] as the number of parameters or dimension increases the search space also increasesexponentially For highly nonlinear problems this dimensionality may be a significantbarrier for almost all optimization algorithms

3 Benchmark Test Functions for Global Optimization

Now we present a collection of 175 unconstrained optimization test problems which canbe used to validate the performance of optimization algorithms The dimensions problemdomain size and optimal solution are denoted by D Lb le xi le Ub and f(xlowast) = f(x1 xn)respectively The symbols Lb and Ub represent lower upper bound of the variables re-spectively It is worth noting that in several cases the optimal solution vectors and theircorresponding solutions are known only as numerical approximations

1 Ackley 1 Function [9](Continuous Differentiable Non-separable Scalable Multi-modal)

f1(x) = minus20eminus002

radic

Dminus1sumD

i=1 x2i minus eD

minus1sumD

i=1 cos(2πxi) + 20 + e

subject to minus35 le xi le 35 The global minima is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

2 Ackley 2 Function [1] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f2(x) = minus200eminus002radic

x21+x2

2

subject to minus32 le xi le 32 The global minimum is located at origin xlowast = (0 0)f(xlowast) = minus200

3 Ackley 3 Function [1] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f3(x) = 200eminus002radic

x21+x2

2 + 5ecos(3x1)+sin(3x2)

subject to minus32 le xi le 32 The global minimum is located at xlowast = (0asymp minus04)f(xlowast) asymp minus2191418

5

4 Ackley 4 or Modified Ackley Function (Continuous Differentiable Non-SeparableScalable Multimodal)

f4(x) =Dsum

i=1

(

eminus02radic

x2i + x2i+1 + 3 (cos(2xi) + sin(2xi+1))

)

subject to minus35 le xi le 35 It is highly multimodal function with two global minimumclose to origin

x = f(minus1479252minus0739807 1479252minus0739807) f(xlowast) = minus3917275

5 Adjiman Function [2](Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f5(x) = cos(x1)sin(x2)minusx1

(x22 + 1)

subject to minus1 le x1 le 2 minus1 le x2 le 1 The global minimum is located at xlowast =(2 010578) f(xlowast) = minus202181

6 Alpine 1 Function [69](Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f6(x) =

Dsum

i=1

∣

∣

∣xisin(xi) + 01xi

∣

∣

∣

subject to minus10 le xi le 10 The global minimum is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

7 Alpine 2 Function [21] (Continuous Differentiable Separable Scalable Multi-modal)

f7(x) =

Dprod

i=1

radicxisin(xi)

subject to 0 le xi le 10 The global minimum is located at xlowast = (7917 middot middot middot 7917)f(xlowast) = 2808D

8 Brad Function [17] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f8(x) =

15sum

i=1

[

yi minus x1 minus uivix2 +wix3

]2

where ui = i vi = 16 minus i wi = min(ui vi) and y = yi = [014 018 022 025 029

032 035 039 037 058 073 096 134 210 439]T It is subject to minus025 le x1 le025 001 le x2 x3 le 25 The global minimum is located at xlowast = (00824 1133 23437)f(xlowast) = 000821487

6

9 Bartels Conn Function (Continuous Non-differentiable Non-Separable Non-ScalableMultimodal)

f9(x) =∣

∣x21 + x22 + x1x2∣

∣+∣

∣sin(x1)∣

∣+∣

∣cos(x2)∣

∣

subject to minus500 le xi le 500 The global minimum is located at xlowast = (0 0) f(xlowast) = 1

10 Beale Function (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f10(x) = (15minus x1 + x1x2)2 + (225 minus x1 + x1x

22)

2

+(2625 minus x1 + x1x32)

2

subject to minus45 le xi le 45 The global minimum is located at xlowast = (3 05) f(xlowast) = 0

11 Biggs EXP2 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f11(x) =

10sum

i=1

(

eminustix1 minus 5eminustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10) f(xlowast) = 0

12 Biggs EXP3 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f12(x) =

10sum

i=1

(

eminustix1 minus x3eminustix2 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 5) f(xlowast) = 0

13 Biggs EXP4 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f13(x) =

10sum

i=1

(

x3eminustix1 minus x4e

minustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 1 5) f(xlowast) = 0

14 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f14(x) =

11sum

i=1

(

x3eminustix1 minus x4e

minustix2 + 3eminustix5 minus yi)2

7

where ti = 01i yi = eminusti minus 5e10ti + 3eminus4ti It is subject to 0 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4) f(xlowast) = 0

15 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f15(x) =

13sum

i=1

(

x3eminustix1 minus x4e

minustix2 + x6eminustix5 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti +3eminus4ti It is subject to minus20 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4 3) f(xlowast) = 0

16 Bird Function [58] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f16(x) = sin(x1)e(1minuscos(x2))2 + cos(x2)e

(1minussin(x1))2 + (x1 minus x2)2

subject to minus2π le xi le 2π The global minimum is located at xlowast = (470104315294)(minus158214 minus313024) f(xlowast) = minus106764537

17 Bohachevsky 1 Function [14] (Continuous Differentiable Separable Non-ScalableMultimodal)

f17(x) = x21 + 2x22 minus 03cos(3πx1)

minus04cos(4πx2) + 07

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

18 Bohachevsky 2 Function [14] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f18(x) = x21 + 2x22 minus 03cos(3πx1) middot 04cos(4πx2)+03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

19 Bohachevsky 3 Function [14] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f19(x) = x21 + 2x22 minus 03cos(3πx1 + 4πx2) + 03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

8

20 Booth Function (Continuous Differentiable Non-separable Non-Scalable Uni-modal)

f20(x) = (x1 + 2x2 minus 7)2 + (2x1 + x2 minus 5)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(1 3) f(xlowast) = 0

21 Box-Betts Quadratic Sum Function [4] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f21(x) =Dminus1sum

i=0

g(xi)2

where

g(x) = eminus01(i+1)x1 minus eminus01(i+1)x2 minus e[(minus01(i+1))minuseminus(i+1) ]x3

subject to 09 le x1 le 12 9 le x2 le 112 09 le x2 le 12 The global minimum islocated at xlowast = f(1 10 1) f(xlowast) = 0

22 Branin RCOS Function [15] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f22(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) + 10

with domain minus5 le x1 le 10 0 le x1 le 15 It has three global minima at xlowast =f(minusπ 12275 π 2275 3π 2425) f(xlowast) = 03978873

23 Branin RCOS 2 Function [60] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f23(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) cos(x2) ln(x21 + x22 + 1) + 10

with domain minus5 le xi le 15 The global minimum is located at xlowast = f(minus32 1253)f(xlowast) = 5559037

24 Brent Function [15] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f24(x) = (x1 + 10)2 + (x2 + 10)2 + eminusx21minusx2

2 (2)

with domain minus10 le xi le 10 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

9

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

4 Ackley 4 or Modified Ackley Function (Continuous Differentiable Non-SeparableScalable Multimodal)

f4(x) =Dsum

i=1

(

eminus02radic

x2i + x2i+1 + 3 (cos(2xi) + sin(2xi+1))

)

subject to minus35 le xi le 35 It is highly multimodal function with two global minimumclose to origin

x = f(minus1479252minus0739807 1479252minus0739807) f(xlowast) = minus3917275

5 Adjiman Function [2](Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f5(x) = cos(x1)sin(x2)minusx1

(x22 + 1)

subject to minus1 le x1 le 2 minus1 le x2 le 1 The global minimum is located at xlowast =(2 010578) f(xlowast) = minus202181

6 Alpine 1 Function [69](Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f6(x) =

Dsum

i=1

∣

∣

∣xisin(xi) + 01xi

∣

∣

∣

subject to minus10 le xi le 10 The global minimum is located at origin xlowast = (0 middot middot middot 0)f(xlowast) = 0

7 Alpine 2 Function [21] (Continuous Differentiable Separable Scalable Multi-modal)

f7(x) =

Dprod

i=1

radicxisin(xi)

subject to 0 le xi le 10 The global minimum is located at xlowast = (7917 middot middot middot 7917)f(xlowast) = 2808D

8 Brad Function [17] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f8(x) =

15sum

i=1

[

yi minus x1 minus uivix2 +wix3

]2

where ui = i vi = 16 minus i wi = min(ui vi) and y = yi = [014 018 022 025 029

032 035 039 037 058 073 096 134 210 439]T It is subject to minus025 le x1 le025 001 le x2 x3 le 25 The global minimum is located at xlowast = (00824 1133 23437)f(xlowast) = 000821487

6

9 Bartels Conn Function (Continuous Non-differentiable Non-Separable Non-ScalableMultimodal)

f9(x) =∣

∣x21 + x22 + x1x2∣

∣+∣

∣sin(x1)∣

∣+∣

∣cos(x2)∣

∣

subject to minus500 le xi le 500 The global minimum is located at xlowast = (0 0) f(xlowast) = 1

10 Beale Function (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f10(x) = (15minus x1 + x1x2)2 + (225 minus x1 + x1x

22)

2

+(2625 minus x1 + x1x32)

2

subject to minus45 le xi le 45 The global minimum is located at xlowast = (3 05) f(xlowast) = 0

11 Biggs EXP2 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f11(x) =

10sum

i=1

(

eminustix1 minus 5eminustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10) f(xlowast) = 0

12 Biggs EXP3 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f12(x) =

10sum

i=1

(

eminustix1 minus x3eminustix2 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 5) f(xlowast) = 0

13 Biggs EXP4 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f13(x) =

10sum

i=1

(

x3eminustix1 minus x4e

minustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 1 5) f(xlowast) = 0

14 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f14(x) =

11sum

i=1

(

x3eminustix1 minus x4e

minustix2 + 3eminustix5 minus yi)2

7

where ti = 01i yi = eminusti minus 5e10ti + 3eminus4ti It is subject to 0 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4) f(xlowast) = 0

15 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f15(x) =

13sum

i=1

(

x3eminustix1 minus x4e

minustix2 + x6eminustix5 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti +3eminus4ti It is subject to minus20 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4 3) f(xlowast) = 0

16 Bird Function [58] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f16(x) = sin(x1)e(1minuscos(x2))2 + cos(x2)e

(1minussin(x1))2 + (x1 minus x2)2

subject to minus2π le xi le 2π The global minimum is located at xlowast = (470104315294)(minus158214 minus313024) f(xlowast) = minus106764537

17 Bohachevsky 1 Function [14] (Continuous Differentiable Separable Non-ScalableMultimodal)

f17(x) = x21 + 2x22 minus 03cos(3πx1)

minus04cos(4πx2) + 07

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

18 Bohachevsky 2 Function [14] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f18(x) = x21 + 2x22 minus 03cos(3πx1) middot 04cos(4πx2)+03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

19 Bohachevsky 3 Function [14] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f19(x) = x21 + 2x22 minus 03cos(3πx1 + 4πx2) + 03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

8

20 Booth Function (Continuous Differentiable Non-separable Non-Scalable Uni-modal)

f20(x) = (x1 + 2x2 minus 7)2 + (2x1 + x2 minus 5)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(1 3) f(xlowast) = 0

21 Box-Betts Quadratic Sum Function [4] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f21(x) =Dminus1sum

i=0

g(xi)2

where

g(x) = eminus01(i+1)x1 minus eminus01(i+1)x2 minus e[(minus01(i+1))minuseminus(i+1) ]x3

subject to 09 le x1 le 12 9 le x2 le 112 09 le x2 le 12 The global minimum islocated at xlowast = f(1 10 1) f(xlowast) = 0

22 Branin RCOS Function [15] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f22(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) + 10

with domain minus5 le x1 le 10 0 le x1 le 15 It has three global minima at xlowast =f(minusπ 12275 π 2275 3π 2425) f(xlowast) = 03978873

23 Branin RCOS 2 Function [60] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f23(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) cos(x2) ln(x21 + x22 + 1) + 10

with domain minus5 le xi le 15 The global minimum is located at xlowast = f(minus32 1253)f(xlowast) = 5559037

24 Brent Function [15] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f24(x) = (x1 + 10)2 + (x2 + 10)2 + eminusx21minusx2

2 (2)

with domain minus10 le xi le 10 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

9

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

9 Bartels Conn Function (Continuous Non-differentiable Non-Separable Non-ScalableMultimodal)

f9(x) =∣

∣x21 + x22 + x1x2∣

∣+∣

∣sin(x1)∣

∣+∣

∣cos(x2)∣

∣

subject to minus500 le xi le 500 The global minimum is located at xlowast = (0 0) f(xlowast) = 1

10 Beale Function (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f10(x) = (15minus x1 + x1x2)2 + (225 minus x1 + x1x

22)

2

+(2625 minus x1 + x1x32)

2

subject to minus45 le xi le 45 The global minimum is located at xlowast = (3 05) f(xlowast) = 0

11 Biggs EXP2 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f11(x) =

10sum

i=1

(

eminustix1 minus 5eminustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10) f(xlowast) = 0

12 Biggs EXP3 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f12(x) =

10sum

i=1

(

eminustix1 minus x3eminustix2 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 5) f(xlowast) = 0

13 Biggs EXP4 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f13(x) =

10sum

i=1

(

x3eminustix1 minus x4e

minustix2 minus yi)2

where ti = 01i yi = eminusti minus 5e10ti It is subject to 0 le xi le 20 The global minimumis located at xlowast = (1 10 1 5) f(xlowast) = 0

14 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f14(x) =

11sum

i=1

(

x3eminustix1 minus x4e

minustix2 + 3eminustix5 minus yi)2

7

where ti = 01i yi = eminusti minus 5e10ti + 3eminus4ti It is subject to 0 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4) f(xlowast) = 0

15 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f15(x) =

13sum

i=1

(

x3eminustix1 minus x4e

minustix2 + x6eminustix5 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti +3eminus4ti It is subject to minus20 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4 3) f(xlowast) = 0

16 Bird Function [58] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f16(x) = sin(x1)e(1minuscos(x2))2 + cos(x2)e

(1minussin(x1))2 + (x1 minus x2)2

subject to minus2π le xi le 2π The global minimum is located at xlowast = (470104315294)(minus158214 minus313024) f(xlowast) = minus106764537

17 Bohachevsky 1 Function [14] (Continuous Differentiable Separable Non-ScalableMultimodal)

f17(x) = x21 + 2x22 minus 03cos(3πx1)

minus04cos(4πx2) + 07

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

18 Bohachevsky 2 Function [14] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f18(x) = x21 + 2x22 minus 03cos(3πx1) middot 04cos(4πx2)+03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

19 Bohachevsky 3 Function [14] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f19(x) = x21 + 2x22 minus 03cos(3πx1 + 4πx2) + 03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

8

20 Booth Function (Continuous Differentiable Non-separable Non-Scalable Uni-modal)

f20(x) = (x1 + 2x2 minus 7)2 + (2x1 + x2 minus 5)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(1 3) f(xlowast) = 0

21 Box-Betts Quadratic Sum Function [4] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f21(x) =Dminus1sum

i=0

g(xi)2

where

g(x) = eminus01(i+1)x1 minus eminus01(i+1)x2 minus e[(minus01(i+1))minuseminus(i+1) ]x3

subject to 09 le x1 le 12 9 le x2 le 112 09 le x2 le 12 The global minimum islocated at xlowast = f(1 10 1) f(xlowast) = 0

22 Branin RCOS Function [15] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f22(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) + 10

with domain minus5 le x1 le 10 0 le x1 le 15 It has three global minima at xlowast =f(minusπ 12275 π 2275 3π 2425) f(xlowast) = 03978873

23 Branin RCOS 2 Function [60] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f23(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) cos(x2) ln(x21 + x22 + 1) + 10

with domain minus5 le xi le 15 The global minimum is located at xlowast = f(minus32 1253)f(xlowast) = 5559037

24 Brent Function [15] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f24(x) = (x1 + 10)2 + (x2 + 10)2 + eminusx21minusx2

2 (2)

with domain minus10 le xi le 10 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

9

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

where ti = 01i yi = eminusti minus 5e10ti + 3eminus4ti It is subject to 0 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4) f(xlowast) = 0

15 Biggs EXP5 Function [13] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f15(x) =

13sum

i=1

(

x3eminustix1 minus x4e

minustix2 + x6eminustix5 minus yi

)2

where ti = 01i yi = eminusti minus 5e10ti +3eminus4ti It is subject to minus20 le xi le 20 The globalminimum is located at xlowast = (1 10 1 5 4 3) f(xlowast) = 0

16 Bird Function [58] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f16(x) = sin(x1)e(1minuscos(x2))2 + cos(x2)e

(1minussin(x1))2 + (x1 minus x2)2

subject to minus2π le xi le 2π The global minimum is located at xlowast = (470104315294)(minus158214 minus313024) f(xlowast) = minus106764537

17 Bohachevsky 1 Function [14] (Continuous Differentiable Separable Non-ScalableMultimodal)

f17(x) = x21 + 2x22 minus 03cos(3πx1)

minus04cos(4πx2) + 07

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

18 Bohachevsky 2 Function [14] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f18(x) = x21 + 2x22 minus 03cos(3πx1) middot 04cos(4πx2)+03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

19 Bohachevsky 3 Function [14] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f19(x) = x21 + 2x22 minus 03cos(3πx1 + 4πx2) + 03

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

8

20 Booth Function (Continuous Differentiable Non-separable Non-Scalable Uni-modal)

f20(x) = (x1 + 2x2 minus 7)2 + (2x1 + x2 minus 5)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(1 3) f(xlowast) = 0

21 Box-Betts Quadratic Sum Function [4] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f21(x) =Dminus1sum

i=0

g(xi)2

where

g(x) = eminus01(i+1)x1 minus eminus01(i+1)x2 minus e[(minus01(i+1))minuseminus(i+1) ]x3

subject to 09 le x1 le 12 9 le x2 le 112 09 le x2 le 12 The global minimum islocated at xlowast = f(1 10 1) f(xlowast) = 0

22 Branin RCOS Function [15] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f22(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) + 10

with domain minus5 le x1 le 10 0 le x1 le 15 It has three global minima at xlowast =f(minusπ 12275 π 2275 3π 2425) f(xlowast) = 03978873

23 Branin RCOS 2 Function [60] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f23(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) cos(x2) ln(x21 + x22 + 1) + 10

with domain minus5 le xi le 15 The global minimum is located at xlowast = f(minus32 1253)f(xlowast) = 5559037

24 Brent Function [15] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f24(x) = (x1 + 10)2 + (x2 + 10)2 + eminusx21minusx2

2 (2)

with domain minus10 le xi le 10 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

9

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

20 Booth Function (Continuous Differentiable Non-separable Non-Scalable Uni-modal)

f20(x) = (x1 + 2x2 minus 7)2 + (2x1 + x2 minus 5)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(1 3) f(xlowast) = 0

21 Box-Betts Quadratic Sum Function [4] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f21(x) =Dminus1sum

i=0

g(xi)2

where

g(x) = eminus01(i+1)x1 minus eminus01(i+1)x2 minus e[(minus01(i+1))minuseminus(i+1) ]x3

subject to 09 le x1 le 12 9 le x2 le 112 09 le x2 le 12 The global minimum islocated at xlowast = f(1 10 1) f(xlowast) = 0

22 Branin RCOS Function [15] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f22(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) + 10

with domain minus5 le x1 le 10 0 le x1 le 15 It has three global minima at xlowast =f(minusπ 12275 π 2275 3π 2425) f(xlowast) = 03978873

23 Branin RCOS 2 Function [60] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f23(x) =

(

x2 minus51x214π2

+5x1π

minus 6

)2

+10

(

1minus 1

8π

)

cos(x1) cos(x2) ln(x21 + x22 + 1) + 10

with domain minus5 le xi le 15 The global minimum is located at xlowast = f(minus32 1253)f(xlowast) = 5559037

24 Brent Function [15] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f24(x) = (x1 + 10)2 + (x2 + 10)2 + eminusx21minusx2

2 (2)

with domain minus10 le xi le 10 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

9

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

25 Brown Function [10] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f25(x) =

nminus1sum

i=1

(x2i )(x2

i+1+1) + (x2i+1)(x2

i+1)

subject to minus1 le xi le 4 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

Bukin functions [80] are almost fractal (with fine seesaw edges) in the surroundingsof their minimal points Due to this property they are extremely difficult to optimizeby any global or local optimization methods

26 Bukin 2 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f26(x) = 100(x2 minus 001x21 + 1) + 001(x1 + 10)2

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

27 Bukin 4 Function (Continuous Non-Differentiable Separable Non-scalable Mul-timodal)

f27(x) = 100x22 + 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 0) f(xlowast) = 0

28 Bukin 6 Function (Continuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f28(x) = 100radic

x2 minus 001x21+ 001x1 + 10

subject to minus15 le x1 le minus5 and minus3 le x2 le minus3 The global minimum is located atxlowast = f(minus10 1) f(xlowast) = 0

29 Camel Function ndash Three Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f29(x) = 2x21 minus 105x41 + x616 + x1x2 + x22

subject to minus5 le xi le 5 The global minima is located at xlowast = f(0 0) f(xlowast) = 0

30 Camel Function ndash Six Hump [15] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f30(x) = (4minus 21x21 +x413)x21

+x1x2 + (4x22 minus 4)x22

10

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

subject tominus5 le xi le 5 The two global minima are located at xlowast = f(minus00898 0712600898minus07126 0) f(xlowast) = minus10316

31 Chen Bird Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f31(x) = minus 0001lfloor

(0001)2 + (x1 minus 04x2 minus 01)2rfloor minus

0001lfloor

(0001)2 + (2x1 + x2 minus 15)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus 718 minus13

18)f(xlowast) = minus2000

32 Chen V Function [19] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f32(x) = minus 0001lfloor

(0001)2 + (x21 + x22 minus 1)2rfloor minus

0001lfloor

(0001)2 + (x21 + x22 minus 05)2rfloor minus

0001lfloor

(0001)2 + (x21 minus x22)2rfloor

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(minus0388888907222222) f(xlowast) = minus2000

33 Chichinadze Function (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f33(x) = x21 minus 12x1 + 11 +

10cos(πx12) + 8sin(5πx12)minus(15)05 exp(minus05(x2 minus 05)2)

subject to minus30 le xi le 30 The global minimum is located at xlowast = f(590133 05)f(xlowast) = minus433159

34 Chung Reynolds Function [20] (Continuous Differentiable Partially-SeparableScalable Unimodal)

f34(x) =(

Dsum

i=1

x2i)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

11

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

35 Cola Function [3] (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

The 17-dimensional function computes indirectly the formula (Du) by setting x0 =y0 x1 = u0 xi = u2(iminus2) yi = u2(iminus2)+1

f35(n u) = h(x y) =sum

jlti

(rij minus dij)2

where rij is given by

rij = [(xi minus xj)2 + (yi minus yj)

2]12

and d is a symmetric matrix given by

d = [dij ] =

127169 143204 235 243309 318 326 285320 322 327 288 155286 256 258 259 312 306317 318 318 312 131 164 300321 318 318 317 170 136 295 132238 231 242 194 285 281 256 291 297

This function has bounds 0 le x0 le 4 and minus4 le xi le 4 for i = 1 D minus 1 It has aglobal minimum of f(xlowast) = 117464

36 Colville Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f36(x) = 100(x1 minus x22)2 + (1minus x1)

2 +

90(x4 minus x23)2 + (1minus x3)

2 +

101((x2 minus 1)2 + (x4 minus 1)2) +

198(x2 minus 1)(x4 minus 1)

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

37 Corana Function [22] (Discontinuous Non-Differentiable Separable Scalable Mul-timodal)

f37(x) =

015(

zi minus 005sgn(zi)2)

di if |vi| lt A

dix2i otherwise

where

vi = |xi minus zi| A = 005

zi = 02lfloor∣

∣

∣

xi02

∣

∣

∣+ 049999

rfloor

sgn (xi)

di = (1 1000 10 100) (-21)

12

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0 0 0)f(xlowast) = 0

38 Cosine Mixture Function [4] (Discontinuous Non-Differentiable Separable Scal-able Multimodal)

f38(x) = minus01

nsum

i=1

cos(5πxi)minusnsum

i=1

x2i

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 0) f(xlowast) =(02 or 04) for n = 2 and 4 respectively

39 Cross-in-Tray Function [58] (Continuous Non-Separable Non-Scalable Multi-modal)

f39(x) = minus00001[|sin(x1)sin(x2)e|100minus[(x2

1+x22)]

05π||+ 1]01

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn1349406685353340 plusmn1349406608602084)f(xlowast) = minus206261218

40 Csendes Function [25] (Continuous Differentiable Separable Scalable Multimodal)

f40(x) =Dsum

i=1

x6i

(

2 + sin1

xi

)

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

41 Cube Function [49] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f41(x) = 100(

x2 minus x31)2

+ (1minus x1)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(minus1 1) f(xlowast) = 0

42 Damavandi Function [26] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f42(x) =

[

1minus∣

∣

∣

∣

sin[π(x1 minus 2)]sin[π(x2 minus 2)]

π2(x1 minus 2)(x2 minus 2)

∣

∣

∣

∣

5]

[

2 + (x1 minus 7)2 + 2(x2 minus 7)2]

subject to 0 le xi le 14 The global minimum is located at xlowast = f(2 2) f(xlowast) = 0

13

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

43 Deb 1 Function (Continuous Differentiable Separable Scalable Multimodal)

f43(x) = minus 1

D

Dsum

i=1

sin6(5πxi)

subject to minus1 le xi le 1 The number of global minima is 5D that are evenly spacedin the function landscape where D represents the dimension of the problem

44 Deb 3 Function (Continuous Differentiable Separable Scalable Multimodal)

f44(x) = minus 1

D

Dsum

i=1

sin6(5π(x34i minus 005))

subject to minus1 le xi le 1 The number of global minima is 5D that are unevenly spacedin the function landscape where D represents the dimension of the problem

45 Deckkers-Aarts Function [4] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f45(x) = 105x21 + x22 minus (x21 + x22)2 + 10minus5(x21 + x22)

4

subject to minus20 le xi le 20 The two global minima are located at xlowast = f(0plusmn15)f(xlowast) = minus24777

46 deVilliers Glasser 1 Function [27](Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f46(x) =

24sum

i=1

[

x1xti2 sin(x3ti + x4)minus yi

]2

where ti = 01(i minus 1) yi = 60137 times 1371ti sin(3112ti + 1761) It is subject tominus500 le xi le 500 The global minimum is f(xlowast) = 0

47 deVilliers Glasser 2 Function [27] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f47(x) =16sum

i=1

[

x1xti2 tanh [x3ti + sin(x4ti)] cos(tie

x5)minus yi]2

where ti = 01(i minus 1) yi = 5381 times 127ti tanh(3012ti + sin(213ti)) cos(e0507ti) It is

subject to minus500 le xi le 500 The global minimum is f(xlowast) = 0

48 Dixon amp Price Function [28] (Continuous Differentiable Non-Separable ScalableUnimodal)

f48(x) = (x1 minus 1)2 +

Dsum

i=2

i(2x2i minus ximinus1)2

14

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(2( 2iminus22i

))f(xlowast) = 0

49 Dolan Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f49(x) = (x1 + 17x2) sin(x1)minus 15x3 minus 01x4 cos(x4 + x5 minus x1) +

02x25 minus x2 minus 1

subject to minus100 le xi le 100 The global minimum is f(xlowast) = 0

50 Easom Function [20](Continuous Differentiable Separable Non-Scalable Multi-modal)

f50(x) = minuscos(x1)cos(x2) exp[minus(x1 minus π)2

minus(x2 minus π)2]

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(π π)f(xlowast) = minus1

51 El-Attar-Vidyasagar-Dutta Function [30] (Continuous Differentiable Non-SeparableNon-Scalable Unimodal)

f51(x) = (x21 + x2 minus 10)2 + (x1 + x22 minus 7)2 +

(x21 + x32 minus 1)2

subject tominus500 le xi le 500 The global minimum is located at xlowast = f(2842503 1920175)f(xlowast) = 0470427

52 Egg Crate Function (Continuous Separable Non-Scalable)

f52(x) = x21 + x22 + 25(sin2(x1) + sin2(x2))

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

53 Egg Holder Function (Continuous Differentiable Non-Separable Scalable Multi-modal)

f53(x) =

mminus1sum

i=1

[minus(xi+1 + 47)sinradic

|xi+1 + xi2 + 47|

minusxisinradic

|xi minus (xi+1 + 47)|]

subject to minus512 le xi le 512 The global minimum is located at xlowast = f(512 4042319)f(xlowast) asymp 95964

15

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

54 Exponential Function [70] (Continuous Differentiable Non-Separable ScalableMultimodal)

f54(x) = minus exp

(

minus05

Dsum

i=1

x2i

)

subject to minus1 le xi le 1 The global minima is located at x = f(0 middot middot middot 0) f(xlowast) = 1

55 Exp 2 Function [3] (Separable)

f55(x) =9sum

i=0

(

eminusix110 minus 5eminusix210 minus eminusi10 + 5eminusi)2

with domain 0 le xi le 20 The global minimum is located at xlowast = f(1 10) f(xlowast) = 0

56 Freudenstein Roth Function [71] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f56(x) = (x1 minus 13 + ((5 minus x2)x2 minus 2)x2)2 +

(x1 minus 29 + ((x2 + 1)x2 minus 14)x2)2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(5 4) f(xlowast) = 0

57 Giunta Function [58] (Continuous Differentiable Separable Scalable Multimodal)

f57(x) = 06 +2sum

i=1

[sin(16

15xi minus 1)

+sin2(16

15xi minus 1)

+1

50sin(4(

16

15xi minus 1))]

subject tominus1 le xi le 1 The global minimum is located at xlowast = f(045834282 045834282)f(xlowast) = 0060447

58 Goldstein Price Function [38] (Continuous Differentiable Non-separable Non-Scalable Multimodal)

f58(x) = [1 + (x1 + x2 + 1)2(19 minus 14x1

+3x21 minus 14x2 + 6x1x2 + 3x22)]

times[30 + (2x1 minus 3x2)2

(18minus 32x1 + 12x21 + 48x2 minus 36x1x2 + 27x22)]

subject to minus2 le xi le 2 The global minimum is located at xlowast = f(0minus1) f(xlowast) = 3

16

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

59 Griewank Function [40] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f59(x) =

nsum

i=1

x2i4000

minusprod

cos(xiradici) + 1

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

60 Gulf Research Problem [79] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f60(x) =99sum

i=1

[

exp

(

minus(ui minus x2)x3

xi

)

minus 001i

]2

where ui = 25 + [minus50 ln(001i)]115 subject to 01 le x1 le 100 0 le x2 le 256 and0 le x1 le 5 The global minimum is located at xlowast = f(50 25 15) f(xlowast) = 0

61 Hansen Function [34] (Continuous Differentiable Separable Non-Scalable Multi-modal)

f61(x) =4sum

i

(i+ 1)cos(ix1 + i+ 1)

4sum

j=0

(j + 1)cos((j + 2)x2 + j + 1)

subject to minus10 le xi le 10 The multiple global minima are located at

xlowast = f(minus7589893minus7708314 minus7589893minus1425128

minus7589893 4858057 minus1306708minus7708314minus1306708 4858057 4976478 4858057 4976478minus1425128 4976478minus7708314)

62 Hartman 3 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f62(x) = minus4sum

i=1

ci exp

minus3sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 2 3 with constants aij pij and ci are given as

17

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

A = [Aij ] =

3 10 3001 10 353 10 3001 10 35

c = ci =

112332

p = pi =

03689 01170 0267304699 04837 0747001091 08732 05547003815 05743 08828

The global minimum is located at xlowast = f(01140 0556 0852) f(xlowast) asymp minus3862782

63 Hartman 6 Function [42] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f63(x) = minus4sum

i=1

ci exp

minus6sum

j=1

aij(xj minus pij)2

subject to 0 le xj le 1 j isin 1 middot middot middot 6 with constants aij pij and ci are given as

A = [Aij ] =

10 3 17 35 17 8005 10 17 01 8 143 35 17 10 17 817 8 005 10 01 14

c = ci =

112332

p = pi =

01312 01696 05569 00124 08283 0558602329 04135 08307 03736 01004 0999102348 01451 03522 02883 03047 0665004047 08828 08732 05743 01091 00381

The global minima is located at x = f(0201690 0150011 0476874 0275332 0311652 0657301) f(xlowast) asymp minus332236

64 Helical Valley [32] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f64(x) = 100

[

(x2 minus 10θ)2 +

(

radic

x21 + x22 minus 1

)]

+x23

where

θ =

12π tan

minus1(

x1x2

)

if x1 ge 0

12π tan

minus1(

x1x2

+ 05)

if x1 lt 0

subject to minus10 le xi le 10 The global minima is located at xlowast = f(1 0 0) f(xlowast) = 0

18

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

65 Himmelblau Function [45] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f65(x) = (x21 + x2 minus 11)2 + (x1 + x22 minus 7)2

subject to minus5 le xi le 5 The global minimum is located at xlowast = f(3 2) f(xlowast) = 0

66 Hosaki Function [11] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f66(x) = (1minus 8x1 + 7x21 minus 73x31 + 14x41)x22e

minusx2

subject to 0 le x1 le 5 and 0 le x2 le 6 The global minimum is located at xlowast = f(4 2)f(xlowast) asymp minus23458

67 Jennrich-Sampson Function [46] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f67(x) =

10sum

i=1

(

2 + 2iminus(

eix1 + eix2))2

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0257825 0257825)f(xlowast) = 1243612

68 Langerman-5 Function [12] (Continuous Differentiable Non-Separable ScalableMultimodal)

f68(x) = minusmsum

i=1

cieminus 1

π

sumDj=1(xjminusaij)

2

cos

π

Dsum

j=1

(xj minus aij)2

subject to 0 le xj le 10 where j isin [0D minus 1] and m = 5 It has a global minimumvalue of f(xlowast) = minus14 The matrix A and column vector c are given as

The matrix A is given by

A = [Aij ] =

9681 0667 4783 9095 3517 9325 6544 0211 5122 20209400 2041 3788 7931 2882 2672 3568 1284 7033 73748025 9152 5114 7621 4564 4711 2996 6126 0734 49822196 0415 5649 6979 9510 9166 6304 6054 9377 14268074 8777 3467 1863 6708 6349 4534 0276 7633 1567

c = ci =

08060517

1509080965

19

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

69 Keane Function (Continuous Differentiable Non-Separable Non-Scalable Multi-modal)

f69(x) =sin2(x1 minus x2)sin

2(x1 + x2)radic

x21 + x22

subject to 0 le xi le 10

The multiple global minima are located at xlowast = f(0 139325139325 0) f(xlowast)=minus0673668

70 Leon Function [49](Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f70(x) = 100(x2 minus x21)2 + (1minus x1)

2

subject to minus12 le xi le 12 A global minimum is located at f(xlowast) = f(1 1)f(xlowast) = 0

71 Matyas Function [43] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f71(x) = 026(x21 + x22)minus 048x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

72 McCormick Function [50] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f72(x) = sin(x1 + x2) + (x1 minus x2)2 minus (32)x1 + (52)x2 + 1

subject to minus15 le x1 le 4 and minus3 le x2 le 3 The global minimum is located atxlowast = f(minus0547minus1547) f(xlowast) asymp minus19133

73 Miele Cantrell Function [24] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f73(x) =(

eminusx1 minus x2)4

+ 100(x2 minus x3)6

+(tan (x3 minus x4))4 + x81

subject to minus1 le xi le 1 The global minimum is located at xlowast = f(0 1 1 1)f(xlowast) = 0

74 Mishra 1 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f74(x) =

(

1 +D minusNminus1sum

i=1

xi

)NminussumNminus1

i=1 xi

(-72)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

20

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

75 Mishra 2 Function [53] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f75(x) =

(

1 +D minusNminus1sum

i=1

05(xi + xi+1)

)NminussumNminus1

i=1 05(xi+xi+1)

(-71)

subject to 0 le xi le 1 The global minimum is f(xlowast) = 2

76 Mishra 3 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f76(x) =

radic

∣

∣cosradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus8466minus10) f(xlowast) = minus018467

77 Mishra 4 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f77(x) =

radic

∣

∣sinradic

∣

∣x21 + x22∣

∣

∣

∣+ 001(x1 + x2)

The global minimum is located at xlowast = f(minus994112minus10) f(xlowast) = minus0199409

78 Mishra 5 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f78(x) =[

sin2(cos((x1) + cos(x2)))2 + cos2(sin(x1) + sin(x2)) + x1

]2

+001(x1 + x2)

The global minimum is located at xlowast = f(minus198682minus10) f(xlowast) = minus101983

79 Mishra 6 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f79(x) = minus ln[

sin2(cos((x1) + cos(x2)))2 minus cos2(sin(x1) + sin(x2)) + x1

]2

+001((x1 minus 1)2 + (x2 minus 1)2)

The global minimum is located at xlowast = f(288631 182326) f(xlowast) = minus228395

80 Mishra 7 Function (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f80(x) =[

Dprod

i=1

xi minusN ]2

The global minimum is f(xlowast) = 0

21

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

81 Mishra 8 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f81(x) = 0001[∣

∣

∣x101 minus 20x91 + 180x81 minus 960x71 + 3360x61 minus 8064x51

1334x41 minus 15360x31 + 11520x21 minus 5120x1 + 2624∣

∣

∣

∣

∣

∣x42 + 12x32 + 54x22 + 108x2 + 81

∣

∣

∣

]2

The global minimum is located at xlowast = f(2minus3) f(xlowast) = 0

82 Mishra 9 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f82(x) =[

ab2c+ abc2 + b2 + (x1 + x2 minus x3)2]2

where a = 2x31 + 5x1x2 + 4x3 minus 2x21x3 minus 18 b = x1 + x32 + x1x23 minus 22

c = 8x21 + 2x2x3 + 2x22 + 3x32 minus 52 The global minimum is located at xlowast = f(1 2 3)f(xlowast) = 0

83 Mishra 10 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f83(x) =[

lfloorx1 perp x2rfloor minus lfloorx1rfloor minus lfloorx2rfloor]2

The global minimum is located at xlowast = f(0 0) (2 2) f(xlowast) = 0

84 Mishra 11 Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f84(x) =[ 1

D

Dsum

i=1

∣

∣xi∣

∣minus(

Dprod

i=1

∣

∣xi∣

∣

)1N

]2

The global minimum is f(xlowast) = 0

85 Parsopoulos Function (Continuous Differentiable Separable Scalable Multimodal)

f85(x) = cos (x1)2 + sin (x2)

2

subject to minus5 le xi le 5 where (x1 x2) isin R2 This function has infinite number of

global minima in R2 at points (κπ

2 λπ) where κ = plusmn1plusmn3 and λ = 0plusmn1plusmn2 In the given domain problem function has 12 global minima all equal to zero

86 Pen Holder Function [58] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f86(x) = minus exp[|cos(x1)cos(x2)e|1minus[(x21+x2

2)]05π||minus1]

subject to minus11 le xi le 11 The four global minima are located at xlowast = f(plusmn9646168plusmn9646168) f(xlowast) = minus096354

22

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

87 Pathological Function [69] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f87(x) =

Dminus1sum

i=1

05 +sin2

radic

100x2i + x2i+1 minus 05

1 + 0001(x2i minus 2xixi+1 + x2i+1)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

88 Paviani Function [45] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f88(x) =

10sum

i=1

[

(ln (xi minus 2))2 + (ln (10minus xi))2]

minus(

10prod

i=1

xi

)02

subject to 20001 le xi le 10 i isin 1 2 10 The global minimum is located atxlowast asymp f(9351 9351) f(xlowast) asymp minus45778

89 Pinter Function [63] (Continuous Differentiable Non-separable Scalable Multi-modal)

f89(x) =

Dsum

i=1

ix2i +

Dsum

i=1

20isin2A+

Dsum

i=1

ilog10(

1 + iB2)

where

A = (ximinus1 sinxi + sinxi+1)

B =(

x2iminus1 minus 2xi + 3xi+1 minus cos xi + 1)

where x0 = xD and xD+1 = x1 subject to minus10 le xi le 10 The global minima islocated at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

90 Periodic Function [4] (Separable)

f90(x) = 1 + sin2(x1) + sin2(x2)minus 01eminus(x21+x2

2)

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 0) f(xlowast) = 09

91 Powell Singular Function [64] (Continuous Differentiable Non-Separable ScalableUnimodal)

f91(x) =

D4sum

i=1

(x4iminus3 + 10x4iminus2)2

+5(x4iminus1 minus x4i)2 + (x4iminus2 minus x4iminus1)

4

+10(x4iminus3 minus x4i)4

23

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

subject tominus4 le xi le 5 The global minima is located at xlowast = f(3minus1 0 1 middot middot middot 3minus1 0 1)f(xlowast) = 0

92 Powell Singular 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f92(x) =Dminus2sum

i=1

(ximinus1 + 10xi)2

+5(xi+1 minus xi+2)2 + (xi minus 2xi+1)

4

+10(ximinus1 minus xi+2)4

subject to minus4 le xi le 5 The global minimum is f(xlowast) = 0

93 Powell Sum Function [69] (Continuous Differentiable Separable Scalable Uni-modal)

f93(x) =Dsum

i=1

∣

∣

∣xi

∣

∣

∣

i+1

subject to minus1 le xi le 1 The global minimum is f(xlowast) = 0

94 Price 1 Function [67] (Continuous Non-Differentiable Separable Non-ScalableMultimodal)

f94(x) = (|x1| minus 5)2 + (|x2| minus 5)2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

95 Price 2 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f95(x) = 1 + sin2 x1 + sin2 x2 minus 01eminusx21minusx2

2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 09

96 Price 3 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f96(x) = 100(x2 minus x21)2 + 6

[

64(x2 minus 05)2 minus x1 minus 06]2

subject tominus500 le xi le 500 The global minimum are located at xlowast =f(minus5minus5minus5 55minus5 5 5) f(xlowast) = 0

24

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

97 Price 4 Function [67] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f97(x) = (2x31x2 minus x32)2 + (6x1 minus x22 + x2)

2

subject tominus500 le xi le 500 The three global minima are located at xlowast = f(0 02 41464minus2506) f(xlowast) = 0

98 Qing Function [68] (Continuous Differentiable Separable Scalable Multimodal)

f98(x) =

Dsum

i=1

(x2i minus i)2

subject to minus500 le xi le 500 The global minima are located at xlowast = f(plusmnradici)

f(xlowast) = 0

99 Quadratic Function (Continuous Differentiable Non-Separable Non-Scalable)

f99(x) = minus380384 minus 13808x1 minus 23292x2

+12808x21 + 20364x22 + 18225x1x2

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(019388 048513)f(xlowast) = minus38737243

100 Quartic Function [81] (Continuous Differentiable Separable Scalable)

f100(x) =

Dsum

i=1

ix4i + random[0 1)

subject to minus128 le xi le 128 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

101 Quintic Function [58](Continuous Differentiable Separable Non-Scalable Multi-modal)

f101(x) =

Dsum

i=1

|x5i minus 3x4i + 4x3i + 2x2i minus 10xi minus 4|

subject to minus10 le xi le 10 The global minimum is located at xlowast = f(-1 or 2)f(xlowast) = 0

102 Rana Function [66] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f102(x) =Dminus2sum

i=0

(xi+1 + 1)cos(t2)sin(t1) + xi lowast cos(t1)sin(t2)

subject to minus500 le xi le 500 where t1 =radic

xi+1 + xi + 1 and t2 =radic

xi+1 minus xi + 1

25

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

103 Ripple 1 Function (Non-separable)

f103(x) =2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi) + 01cos2(500πxi))

subject to 0 le xi le 1 It has one global minimum and 252004 local minima Theglobal form of the function consists of 25 holes which forms a 5 times 5 regular gridAdditionally the whole function landscape is full of small ripples caused by highfrequency cosine function which creates a large number of local minima

104 Ripple 25 Function (Non-separable)

f104(x) =

2sum

i=1

minuse-2 ln2(ximinus01

08)2(sin6(5πxi))

subject to 0 le xi le 1 It has one global form of the Ripple-1 function without anyripples due to absence of cosine term

105 Rosenbrock Function [73] (Continuous Differentiable Non-Separable ScalableUnimodal)

f105(x) =Dminus1sum

i=1

[

100(xi+1 minus x2i )2 + (xi minus 1)2

]

subject to minus30 le xi le 30 The global minima is located at xlowast = f(1 middot middot middot 1)f(xlowast) = 0

106 Rosenbrock Modified Function (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f106(x) = 74 + 100(x2 minus x21)2 + (1minus x)2

minus400eminus(x1+1)2+(x2+1)2

01

subject to minus2 le xi le 2 In this function a Gaussian bump at (minus1 1) is added whichcauses a local minimum at (1 1) and global minimum is located at xlowast = f(minus1minus1)f(xlowast) = 0 This modification makes it a difficult to optimize because local minimumbasin is larger than the global minimum basin

107 Rotated Ellipse Function (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f107(x) = 7x21 minus 6radic3x1x2 + 13x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

26

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

108 Rotated Ellipse 2 Function [66] (Continuous Differentiable Non-Separable Non-Scalable Unimodal)

f108(x) = x21 minus x1x2 + x22

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0) f(xlowast) =0s

109 Rump Function [51] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f109(x) = (33375 minus x21)x62 + x21(11x

21x

22 minus 121x42 minus 2) + 55x82 +

x12x2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

110 Salomon Function [74] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f110(x) = = 1minus cos

(

2π

radic

radic

radic

radic

Dsum

i=1

x2i

)

+ 01

radic

radic

radic

radic

Dsum

i=1

x2i

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

111 Sargan Function [29] (Continuous Differentiable Non-Separable Scalable Multi-modal)

f111(x) = =sum

i=1

D(

x2i + 04sum

j 6=1

xixj

)

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

112 Scahffer 1 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f112(x) = 05 +sin2(x21 + x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

27

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

113 Scahffer 2 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f113(x) = 05 +sin2(x21 minus x22)

2 minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 0)f(xlowast) = 0

114 Scahffer 3 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f114(x) = 05 +sin2

(

cos∣

∣

∣x21 minus x22

∣

∣

∣

)

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 000156685

115 Scahffer 4 Function [59] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f115(x) = 05 +cos2

(

sin(x21 minus x22))

minus 05

1 + 0001(x21 + x22)2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 1253115)f(xlowast) = 0292579

116 Schmidt Vetters Function [50] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f116(x) =1

1 + (x1 minus x2)2+ sin

(πx2 + x32

)

+e(x1+x2

x2minus2)2

The global minimum is located at xlowast = f(078547 078547 078547) f(xlowast) = 3

117 Schumer Steiglitz Function [75] (Continuous Differentiable Separable ScalableUnimodal)

f117(x) =

Dsum

i=1

x4i

The global minimum is located at xlowast = f(0 0) f(xlowast) = 0

28

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

118 Schwefel Function [77] (Continuous Differentiable Partially-Separable ScalableUnimodal)

f118(x) =(

Dsum

i=1

x2i

)α

where α ge 0 subject to minus100 le xi le 100 The global minima is located atxlowast = f(0 middot middot middot 0) f(xlowast) = 0

119 Schwefel 12 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f119(x) =

Dsum

i=1

isum

j=1

xj

2

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

120 Schwefel 24 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f120(x) =Dsum

i=1

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

121 Schwefel 26 Function [77] (Continuous Differentiable Non-Separable Non-ScalableUnimodal)

f121(x) = max(|x1 + 2x2 minus 7| |2x1 + x2 minus 5|)

subject to minus100 le xi le 100 The global minima is located at xlowast = f(1 3) f(xlowast) = 0

122 Schwefel 220 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f122(x) = minusnsum

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

123 Schwefel 221 Function [77] (Continuous Non-Differentiable Separable ScalableUnimodal)

f123(x) = max1leileD

|xi|

29

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

124 Schwefel 222 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f124(x) =

Dsum

i=1

|xi|+nprod

i=1

|xi|

subject to minus100 le xi le 100 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

125 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f125(x) =

Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

126 Schwefel 223 Function [77] (Continuous Differentiable Non-Separable ScalableUnimodal)

f126(x) =Dsum

i=1

x10i

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

127 Schwefel 225 Function [77] (Continuous Differentiable Separable Non-ScalableMultimodal)

f127(x) =

Dsum

i=2

(xi minus 1)2 + (x1 minus x2i )2

subject to 0 le xi le 10 The global minima is located at xlowast = f(1 middot middot middot 1) f(xlowast) = 0

128 Schwefel 226 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f128(x) = minus 1

D

Dsum

i=1

xi sinradic

|xi|

subject to minus500 le xi le 500 The global minimum is located at xlowast = plusmn[π(05 + k)]2f(xlowast) = minus418983

30

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

129 Schwefel 236 Function [77] (Continuous Differentiable Separable Scalable Mul-timodal)

f129(x) = minusx1x2(72 minus 2x1 minus 2x2)

subject to 0 le xi le 500 The global minimum is located at xlowast = f(12 middot middot middot 12)f(xlowast) = minus3456

130 Shekel 5 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f130(x) = minus5sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 7

c = ci =

0102020404

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus101499

131 Shekel 7 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f131(x) = minus7sum

i=1

14sum

j=1(xj minus aij)

2 + ci

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 3

c = ci =

01020204040603

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus103999

132 Shekel 10 [62] (Continuous Differentiable Non-Separable Scalable Multimodal)

f132(x) = minus10sum

i=1

14sum

j=1(xj minus aij)

2 + ci

31

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

where A = [Aij ] =

4 4 4 41 1 1 18 8 8 86 6 6 63 7 3 72 9 2 95 5 3 38 1 8 16 2 6 27 36 7 36

c = ci =

01020204040603070505

subject to 0 le xj le 10 The global minima is located at xlowast = f(4 4 4 4) f(xlowast) asympminus105319

133 Shubert Function [44] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f133(x) =

nprod

i=1

5sum

j=1

cos((j + 1)xi + j)

subject to minus10 le xi le 10 i isin 1 2 middot middot middot n The 18 global minima are located atxlowast = f(minus70835 48580 minus70835minus77083

minus14251minus70835 54828 48580minus14251minus08003 48580 54828minus77083minus70835 minus70835minus14251minus77083minus08003 minus77083 54828minus08003minus77083 minus08003minus14251minus08003 48580 minus14251 54828 54828minus77083 48580minus70835 54828minus14251 48580minus08003)

f(xlowast) ≃ minus1867309

134 Shubert 3 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f134(x) =

Dsum

i=1

5sum

j=1

jsin((j + 1)xi + j)

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus296733337 with multiplesolutions

135 Shubert 4 Function [3] (Continuous Differentiable Separable Non-Scalable Mul-timodal)

f135(x) =

Dsum

i=1

5sum

j=1

jcos((j + 1)xi + j)

32

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

subject to minus10 le xi le 10 The global minimum is f(xlowast) ≃ minus25740858 with multiplesolutions

136 Schaffer F6 Function [76] (Continuous Differentiable Non-Separable ScalableMultimodal)

f136(x) =Dsum

i=1

05 +sin2

radic

x2i + x2i+1 minus 05[

1 + 0001(x2i + x2i+1)]2

subject to minus100 le xi le 100 The global minimum is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

137 Sphere Function [75] (Continuous Differentiable Separable Scalable Multimodal)

f137(x) =

Dsum

i=1

x2i

subject to 0 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

138 Step Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f138(x) =

Dsum

i=1

(lfloor|xi|rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

139 Step 2 Function [9] (Discontinuous Non-Differentiable Separable Scalable Uni-modal)

f139(x) =Dsum

i=1

(lfloorxi + 05rfloor)2

subject to minus100 le xi le 100 The global minima is located xlowast = f(05 middot middot middot 05) = 0f(xlowast) = 0

140 Step 3 Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f140(x) =

Dsum

i=1

(

lfloorx2i rfloor)

subject to minus100 le xi le 100 The global minima is located xlowast = f(0 middot middot middot 0) = 0f(xlowast) = 0

33

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

141 Stepint Function (Discontinuous Non-Differentiable Separable Scalable Unimodal)

f141(x) = 25 +Dsum

i=1

(lfloorxirfloor)

subject to minus512 le xi le 512 The global minima is located xlowast = f(0 middot middot middot 0)f(xlowast) = 0

142 Streched V Sine Wave Function [76] (Continuous Differentiable Non-SeparableScalable Unimodal)

f142(x) =

Dminus1sum

i=1

(x2i+1 + x2i )025[

sin250(x2i+1 + x2i )01+ 01

]

subject to minus10 le xi le 10 The global minimum is located xlowast = f(0 0) f(xlowast) = 0

143 Sum Squares Function [43] (Continuous Differentiable Separable Scalable Uni-modal)

f143(x) =

Dsum

i=1

ix2i

subject to minus10 le xi le 10 The global minima is located xlowast = f(0 middot middot middot 0) f(xlowast) = 0

144 Styblinski-Tang Function [80] (Continuous Differentiable Non-Separable Non-Scalable Multimodal)

f144(x) =1

2

nsum

i=1

(x4i minus 16x2i + 5xi)

subject tominus5 le xi le 5 The global minimum is located xlowast = f(minus2903534minus2903534)f(xlowast) = minus78332

145 Table 1 Holder Table 1 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f145(x) = minus|cos(x1)cos(x2)e|1minus(x1+x2)05π||

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646168 plusmn9646168) f(xlowast) =minus26920336

146 Table 2 Holder Table 2 Function [58] (Continuous Differentiable SeparableNon-Scalable Multimodal)

f146(x) = minus|sin(x1)cos(x2)e|1minus(x1+x2)05π||

34

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn8055023472141116 plusmn9664590028909654)f(xlowast) = minus1920850

147 Table 3 Carrom Table Function [58] (Continuous Differentiable Non-SeparableNon-Scalable Multimodal)

f147(x) = minus[(cos(x1)cos(x2)

exp |1minus [(x21 + x22)05]π|)2]30

subject to minus10 le xi le 10

The four global minima are located at xlowast = f(plusmn9646157266348881 plusmn9646134286497169)f(xlowast) = minus241568155

148 Testtube Holder Function [58] (Continuous Differentiable Separable Non-ScalableMultimodal)

f148(x) = minus4[

(sin(x1)cos(x2)

e|cos[(x21+x2

2)200]|)]

subject to minus10 le xi le 10 The two global minima are located at xlowast = f(plusmnπ2 0)f(xlowast) = minus10872300

149 Trecanni Function [29] (Continuous Differentiable Separable Non-Scalable Uni-modal)

f149(x) = x41 minus 4x31 + 4x1 + x22

subject to minus5 le xi le 5 The two global minima are located at xlowast = f(0 0 minus2 0)f(xlowast) = 0

150 Trid 6 Function [43] (Continuous Differentiable Non-Separable Non-Scalable Mul-timodal)

f150(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus62 le xi le 62 The global minima is located at f(xlowast) = minus50

151 Trid 10 Function [43] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f151(x) =Dsum

i=1

(xi minus 1)2 minusDsum

i=1

xiximinus1

subject to minus100 le xi le 100 The global minima is located at f(xlowast) = minus200

35

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

152 Trefethen Function [3] (Continuous Differentiable Non-Separable Non-ScalableMultimodal)

f152(x) = esin(50x1) + sin(60ex2)

+sin(70sin(x1)) + sin(sin(80x2))

minussin(10(x1 + x2)) +1

4(x21 + x22)

subject tominus10 le xi le 10 The global minimum is located at xlowast = f(minus0024403 0210612)f(xlowast) = minus330686865

153 Trigonometric 1 Function [29] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f153(x) =

Dsum

i=1

[D minusDsum

j=1

cos xj

+i(1minus cos(xi)minus sin(xi))]2

subject to 0 le xi le pi The global minimum is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

154 Trigonometric 2 Function [35] (Continuous Differentiable Non-Separable Scal-able Multimodal)

f154(x) = 1 +

Dsum

i=1

8 sin2[

7(xi minus 09)2]

+ 6 sin2[

14(x1 minus 09)2]

+ (xi minus 09)2

subject to minus500 le xi le 500 The global minimum is located at xlowast = f(09 middot middot middot 09)f(xlowast) = 1

155 Tripod Function [69] (Discontinuous Non-Differentiable Non-Separable Non-ScalableMultimodal)

f155(x) = p(x2)(1 + p(x1))

+|x1 + 50p(x2)(1minus 2p(x1))|+|x2 + 50(1 minus 2p(x2))|

subject to minus100 le xi le 100 where p(x) = 1 for x ge 0 The global minimum is locatedat xlowast = f(0minus50) f(xlowast) = 0

156 Ursem 1 Function [72] (Separable)

f156(x) = minussin(2x1 minus 05π) minus 3cos(x2)minus 05x1

subject to minus25 le x1 le 3 and minus2 le x2 le 2 and has single global and local minima

36

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

157 Ursem 3 Function [72](Non-separable)

f157(x) = minussin(22πx1 + 05π)2 minus |x1|

23minus |x1|

2

minussin(05πx22 + 05π)2 minus |x2|

23minus |x2|

2

subject to minus2 le x1 le 2 and minus15 le x2 le 15 and has single global minimum and fourregularly spaced local minima positioned in a direct line such that global minimumis in the middle

158 Ursem 4 Function [72] (Non-separable)

f158(x) = minus3sin(05πx1 + 05π)2 minus

radic

x21 + x224

subject to minus2 le xi le 2 and has single global minimum positioned at the middle andfour local minima at the corners of the search space

159 Ursem Waves Function [72](Non-separable)

f159(x) = minus09x21 + (x22 minus 45x22)x1x2

+47cos(3x1 minus x22(2 + x1))sin(25πx1)

subject to minus09 le x1 le 12 and minus12 le x2 le 12 and has single global minimum andnine irregularly spaced local minima in the search space

160 Venter Sobiezcczanski-Sobieski Function [10] (Continuous Differentiable Sep-arable Non-Scalable)

f160(x) = x21 minus 100cos(x1)2

minus100cos(x2130) + x22

minus100cos(x2)2 minus 100cos(x2230)

subject to minus50 le xi le 50 The global minimum is located at xlowast = f(0 0) f(xlowast) =minus400

161 Watson Function [77] (Continuous Differentiable Non-Separable Scalable Uni-modal)

f161(x) =

29sum

i=0

4sum

j=0

((j minus 1)ajixj+1)minus

5sum

j=0

ajixj+1

2

minus 12 + x21

subject to |xi| le 10 where the coefficient ai = i290 The global minimum is locatedat xlowast = f(minus00158 1012minus02329 1260minus1513 09928) f(xlowast) = 0002288

37

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

162 Wayburn Seader 1 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f162(x) = (x61 + x42 minus 17)2 + (2x1 + x2 minus 4)2

The global minimum is located at xlowast = f(1 2) (1597 0806) f(xlowast) = 0

163 Wayburn Seader 2 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f163(x) =[

1613 minus 4(x1 minus 03125)2 minus 4(x2 minus 1625)2]2

+ (x2 minus 1)2

subject to minus500 le 500 The global minimum is located at xlowast = f(02 1) (0425 1)f(xlowast) = 0

164 Wayburn Seader 3 Function [85] (Continuous Differentiable Non-Separable Scal-able Unimodal)

f164(x) = 2x313

minus 8x21 + 33x1 minus x1x2 + 5 +[

(x1 minus 4)2 + (x2 minus 5)2 minus 4]2

subject to minus500 le 500 The global minimum is located at xlowast = f(5611 6187)f(xlowast) = 2135

165 W Wavy Function [23] (Continuous Differentiable Separable Scalable Multi-modal)

f165(x) = 1minus 1

D

Dsum

i=1

cos(kxi)eminusx2i2

subject to minusπ le xi le π The global minimum is located at xlowast = f(0 0) f(xlowast) = 0The number of local minima is kn and (k + 1)n for odd and even k respectively ForD = 2 and k = 10 there are 121 local minima

166 Weierstrass Function [82](Continuous Differentiable Separable Scalable Multi-modal)

f166(x) =nsum

i=1

[kmaxsum

k=0

akcos(2πbk(xi + 05))

minusn

kmaxsum

k=0

akcos(πbk)]

subject to minus05 le xi le 05 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

38

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

167 Whitley Function [86] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f167(x) =Dsum

i=1

Dsum

j=1

[(100(x2i minus xj)2 + (1minus xj)

2)2

4000

minuscos(

100(x2i minus xj)2 + (1minus xj)

2 + 1) ]

combines a very steep overall slope with a highly multimodal area around the globalminimum located at xi = 1 where i = 1 D

168 Wolfe Function [77] (Continuous Differentiable Separable Scalable Multimodal)

f168(x) =4

3(x21 + x22 minus x1x2)

075 + x3 (-199)

subject to 0 le xi le 2 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

169 Xin-She Yang (Function 1) (Separable)This is a generic stochastic and non-smooth function proposed in [88 ]

f169(x) =Dsum

i=1

ǫi|xi|i

subject to minus5 le xi le 5 The variable ǫi (i = 1 2 middot middot middot D) is a random variableuniformly distributed in [0 1] The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

170 Xin-She Yang (Function 2)(Non-separable)

f170(x) =(

Dsum

i=1

|xi|)

exp[

minusDsum

i=1

sin(x2i )]

subject to minus2π le xi le 2π The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = 0

171 Xin-She Yang (Function 3) (Non-separable)

f171(x) =

[

eminussumD

i=1(xiβ)2m minus 2eminussumD

i=1(xi)2

Dprod

i=1

cos2(xi)

]

subject to minus20 le xi le 20 The global minima for m = 5 and β = 15 is located atxlowast = f(0 middot middot middot 0) f(xlowast) = minus1

39

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

172 Xin-She Yang (Function 4) (Non-separable)

f172(x) =

[

Dsum

i=1

sin2(xi)minus eminussumD

i=1 x2i

]

eminussumD

i=1 sin2radic

|xi|

subject to minus10 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0)f(xlowast) = minus1

173 Zakharov Function [69] (Continuous Differentiable Non-Separable Scalable Mul-timodal)

f173(x) =nsum

i=1

x2i +

(

1

2

nsum

i=1

ixi

)2

+

(

1

2

nsum

i=1

ixi

)4

subject to minus5 le xi le 10 The global minima is located at xlowast = f(0 middot middot middot 0) f(xlowast) = 0

174 Zettl Function [78] (Continuous Differentiable Non-Separable Non-Scalable Uni-modal)

f174(x) = (x21 + x22 minus 2x1)2 + 025x1

subject to minus5 le xi le 10 The global minima is located at xlowast = f(minus00299 0)f(xlowast) = minus0003791

175 Zirilli or Aluffi-Pentinirsquos Function [4] (Continuous Differentiable SeparableNon-Scalable Unimodal)

f175(x) = 025x41 minus 05x21 + 01x1 + 05x22

subject to minus10 le xi le 10 The global minimum is located at xlowast = (minus10465 0)f(xlowast) asymp minus03523

4 Conclusions

Test functions are important to validate and compare optimization algorithms especiallynewly developed algorithms Here we have attempted to provide the most comprehensivelist of known benchmarks or test functions However it is may be possibly that we havemissed some functions but this is not intentional This list is based on all the literatureknown to us by the time of writing It can be expected that all these functions should beused for testing new optimization algorithms so as to provide a more complete view aboutthe performance of any algorithms of interest

40

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

References

[1] D H Ackley ldquoA Connectionist Machine for Genetic Hill-Climbingrdquo Kluwer 1987

[2] C S Adjiman S Sallwig C A Flouda A Neumaier ldquoA Global OptimizationMethod aBB for General Twice-Differentiable NLPs-1 Theoretical Advancesrdquo Com-puters Chemical Engineering vol 22 no 9 pp 1137-1158 1998

[3] E P Adorio U P Dilman ldquoMVF - Multivariate Test Function Libraryin C for Unconstrained Global Optimization Methodsrdquo [Available Online]httpwwwgeocitieswseadoriomvfpdf

[4] M M Ali C Khompatraporn Z B Zabinsky ldquoA Numerical Evaluation of SeveralStochastic Algorithms on Selected Continuous Global Optimization Test ProblemsrdquoJournal of Global Optimization vol 31 pp 635-672 2005

[5] N Andrei ldquoAn Unconstrained Optimization Test Functions Collectionrdquo AdvancedModeling and Optimization vol 10 no 1 pp147-161 2008

[6] A Auger N Hansen N Mauny R Ros M Schoenauer ldquoBio-Inspired ContinuousOptimization The Coming of Agerdquo Invited Lecture IEEE Congress on EvolutionaryComputation NJ USA 2007

[7] B M Averick R G Carter J J More ldquoThe MINIPACK-2 Test Problem CollectionrdquoMathematics and Computer Science Division Agronne National Laboratory TechnicalMemorandum No 150 1991

[8] B M Averick R G Carter J J More G L Xue ldquoThe MINIPACK-2 Test ProblemCollectionrdquo Mathematics and Computer Science Division Agronne National Labora-tory Preprint MCS-P153-0692 1992

[9] T Back H P Schwefel ldquoAn Overview of Evolutionary Algorithm for Parameter Op-timizationrdquo Evolutionary Computation vol 1 no 1 pp 1-23 1993

[10] O Begambre J E Laier ldquoA hybrid Particle Swarm Optimization - Simplex Algorithm(PSOS) for Structural Damage Identificationrdquo Journal of Advances in EngineeringSoftware vol 40 no 9 pp 883-891 2009

[11] G A Bekey M T Ung ldquoA Comparative Evaluation of Two Global Search Algo-rithmsrdquo IEEE Transaction on Systems Man and Cybernetics vol 4 no 1 pp 112-116 1974

[12] H Bersini M Dorigo S Langerman ldquoResults of the First International Contest onEvolutionary Optimizationrdquo IEEE International Conf on Evolutionary ComputationNagoya Japan pp 611-615 1996

[13] M C Biggs ldquoA New Variable Metric Technique Taking Account of Non-QuadraticBehaviour of the Objective functionrdquo IMA Journal of Applied Mathematics vol 8no 3 pp 315-327 1971

[14] I O Bohachevsky M E Johnson M L Stein ldquoGeneral Simulated Annealing forFunction Optimizationrdquo Technometrics vol 28 no 3 pp 209-217 1986

41

[15] F H Branin Jr ldquoWidely Convergent Method of Finding Multiple Solutions of Simul-taneous Nonlinear Equationsrdquo IBM Journal of Research and Development vol 16 no5 pp 504-522 1972

[16] D O Boyer C H Martfnez N G Pedrajas rdquoCrossover Operator for EvolutionaryAlgorithms Based on Population Featuresrdquo Journal of Artificial Intelligence Researchvol 24 pp 1-48 2005

[17] Y Brad ldquoComparison of Gradient Methods for the Solution of Nonlinear ParametricEstimation Problemrdquo SIAM Journal on Numerical Analysis vol 7 no 1 pp 157-1861970

[18] The Cross-Entropy ToolboxhttpwwwmathsuqeduauCEToolBox

[19] Y Chen ldquoComputer Simulation of Electron Positron Annihila-tion Processesrdquo Technical Report SLAC-Report-646 Stanford Lin-ear Accelerator Center Stanford University 2003 [Available Online]httpwwwslacstanfordedupubsslacreportsslac-r-646html

[20] C J Chung R G Reynolds ldquoCAEP An Evolution-Based Tool for Real-Valued Func-tion Optimization Using Cultural Algorithmsrdquo International Journal on Artificial In-telligence Tool vol 7 no 3 pp 239-291 1998

[21] M Clerc ldquoThe Swarm and the Queen Towards a Deterministic and Adaptive ParticleSwarm Optimization rdquo IEEE Congress on Evolutionary Computation WashingtonDC USA pp 1951-1957 1999

[22] A Corana M Marchesi C Martini S Ridella ldquoMinimizing Multimodal Functions ofContinuous Variables with Simulated Annealing Algorithmsrdquo ACM Transactions onMathematical Software vol 13 no 3 pp 262-280 1987

[23] P courrieu ldquoThe Hyperbell Algorithm for Global Optimization A Random WalkUsing Cauchy Densitiesrdquo Journal of Global Optimization vol 10 no 1 pp 111-1331997

[24] E E Cragg A V Levy ldquoStudy on Supermemory Gradient Method for the Minimiza-tion of Functionsrdquo Journal of Optimization Theory and Applications vol 4 no 3 pp191-205 1969

[25] T Csendes D Ratz ldquoSubdivision Direction Selection in Interval Methods for GlobalOptimizationrdquo SIAM Journal on Numerical Analysis vol 34 no 3 pp 922-938

[26] N Damavandi S Safavi-Naeini ldquoA Hybrid Evolutionary Programming Method forCircuit Optimizationrdquo IEEE Transaction on Circuit and Systems I vol 52 no 5 pp902-910 2005

[27] N deVillers D Glasser ldquoA Continuation Method for Nonlinear Regressionrdquo SIAMJournal on Numerical Analysis vol 18 no 6 pp 1139-1154 1981

[28] L C W Dixon R C Price ldquoThe Truncated Newton Method for Sparse UnconstrainedOptimisation Using Automatic Differentiationrdquo Journal of Optimization Theory andApplications vol 60 no 2 pp 261-275 1989

42

[29] L C W Dixon G P Szego (eds) ldquoTowards Global Optimization 2rdquo Elsevier 1978

[30] R A El-Attar M Vidyasagar S R K Dutta ldquoAn Algorithm for II-norm Minimiza-tion With Application to Nonlinear II-approximationrdquo SIAM Journal on NumvericalAnalysis vol 16 no 1 pp 70-86 1979

[31] ldquoTwo Algorithms for Global Optimization of General NLP Problemsrdquo InternationalJournal on Numerical Methods in Engineering vol 39 no 19 pp 3305-3325 1996

[32] R Fletcher M J D Powell ldquoA Rapidly Convergent Descent Method for Min-imzationrdquo Computer Journal vol 62 no 2 pp 163-168 1963 [Available Online]httpgaltonuchicagoedu~lekhengcourses302classicsfletcher-powellpdf

[33] C A Flouda P M Pardalos C S Adjiman W R Esposito Z H Gumus S THarding J L Klepeis C A Meyer and C A Schweiger ldquoHandbook of Test Problemsin Local and Global Optimizationrdquo Kluwer Boston 1999

[34] C Fraley ldquoSoftware Performances on Nonlinear Least-Squares Prob-lemsrdquo Technical Report no STAN-CS-89-1244 Department ofComputer Science Stanford University 1989 [Available Online]httpwwwdticmildtictrfulltextu2a204526pdf

[35] M C Fu J Hu S I Marcus ldquoModel-Based Randomized Methods for Global Opti-mizationrdquo Proc 17th International Symp Mathematical Theory Networks SystemsKyoto Japan pp 355-365 2006

[36] GAMS World GLOBAL Library [Available Online]httpwwwgamsworldorgglobalgloballibhtml

[37] GEATbx - The Genetic and Evolutionary Algorithm Toolbox for Matlab [AvailableOnline] httpwwwgeatbxcom

[38] A A Goldstein J F Price ldquoOn Descent from Local Minimardquo Mathematics andComptutaion vol 25 no 115 pp 569-574 1971

[39] V S Gordon D Whitley Serial and Parallel Genetic Algoritms as Function Optimiz-ersrdquo In S Forrest (Eds) 5th Intl Conf on Genetic Algorithms pp 177-183 MorganKaufmann

[40] A O Griewank ldquoGeneralized Descent for Global Optimizationrdquo Journal of Optimiza-tion Theory and Applications vol 34 no 1 pp 11-39 1981

[41] N I M Gould D Orban and P L Toint ldquoCUTEr A Constrainedand Un-constrained Testing Environment Revisitedrdquo [Available Online]httpcuterrlacukcuter-wwwproblemshtml

[42] J K Hartman ldquoSome Experiments in Global Optimizationrdquo [Available Online]httpia701505usarchiveorg9itemssomeexperimentsi00hartsomeexperimentsi00hartpdf

[43] A-R Hedar ldquoGlobal Optimization Test Problemsrdquo [Available Online]httpwww-optimaampikyoto-uacjpmemberstudenthedarHedar_filesTestGOhtm

[44] J P Hennart (ed) ldquoNumerical Analysisrdquo Proc 3rd AS Workshop Lecture Notes inMathematics vol 90 Springer 1982

43

[45] D M Himmelblau ldquoApplied Nonlinear Programmingrdquo McGraw-Hill 1972

[46] R I Jennrich P F Sampson ldquoApplication of Stepwise Regression toNon-Linear estimationrdquo Techometrics vol 10 no 1 pp 63-72 1968httpwwwjstororgdiscover1023071266224uid=3737864ampuid=2129ampuid=2ampuid=70ampuid=4ampsid=211

[47] A D Junior R S Silva K C Mundim and L E Dardenne ldquoPerformanceand Parameterization of the Algorithm Simplified Generalized Simulated An-nealingrdquo Genet Mol Biol vol 27 no 4 pp 616ndash622 2004 [Available Online]httpwwwscielobrscielophpscript=sci_arttextamppid=S1415-47572004000400024amplng=enampnrm=iso

ISSN 1415-4757 httpdxdoiorg101590S1415-47572004000400024

[48] Test Problems for Global Optimization httpwww2immdtudk~kajmTest_ex_formstest_exhtml

[49] A Lavi T P Vogel (eds) ldquoRecent Advances in Optimization Techniquesrdquo JohnWlileyamp Sons 1966

[50] F A Lootsma (ed) ldquo Numerical Methods for Non-Linear Optimizationrdquo AcademicPress 1972

[51] R E Moore ldquoReliability in Computingrdquo Academic Press 1998

[52] J J More B S Garbow K E Hillstrom ldquoTesting Unconstrained Optimization Soft-warerdquo ACM Trans on Mathematical Software vol 7 pp 17-41 1981

[53] S K Mishra ldquoPerformance of Differential Evolution and Particle Swarm Methodson Some Relatively Harder Multi-modal Benchmark Functionsrdquo [Available Online]httpmpraubuni-muenchende449

[54] S K Mishra ldquoPerformance of the Barter the Differential Evolution and the Simu-lated Annealing Methods of Global Pptimization On Some New and Some Old TestFunctionsrdquo [Available Online] httpwwwssrncomabstract=941630

[55] S K Mishra ldquoRepulsive Particle Swarm Method On Some Difficult Test Problems ofGlobal Optimizationrdquo [Available Online] httpmpraubuni-muenchende1742

[56] S K Mishra ldquoPerformance of Repulsive Particle Swarm Method in Global Opti-mization of Some Important Test Functions A Fortran Programrdquo [Available Online]httpwwwssrncomabstract=924339

[57] S K Mishra ldquoGlobal Optimization by Particle Swarm Method A For-tran Programrdquo Munich Research Papers in Economics [Available Online]httpmpraubuni-muenchende874

[58] S K Mishra ldquoGlobal Optimization By Differential Evolution and Particle SwarmMethods Evaluation On Some Benchmark Functionsrdquo Munich Research Papers inEconomics [Available Online] httpmpraubuni-muenchende1005

[59] S K Mishra ldquoSome New Test Functions For Global Optimization AndPerformance of Repulsive Particle Swarm Methodrdquo [Available Online]httpmpraubuni-muenchende2718

44

[60] C Muntenau V Lazarescu ldquoGlobal Search Using a New Evo-lutionayr Framework The Adaptive Reservoir Genetic Algo-rithmrdquo Complexity International vol 5 1998 [Available Online]httpwwwcomplexityorgaucivol05munteanumunteanuhtml

[61] A Neumaier ldquoCOCONUT Benchmarkrdquo [Available Online]httpwwwmatunivieacat~neumgloptcoconutbenchmarkhtml

[62] J Opacic ldquoA Heuristic Method for Finding Most extrema of a Nonlinear FunctionalrdquoIEEE Transactions on Systems Man and Cybernetics vol 3 no 1 pp 102-107 1973

[63] J D Pinter ldquoGlobal Optimization in Action Continuous and Lipschitz OptimizationAlgorithms Implementations and Applicationsrdquo Kluwer 1996

[64] M J D Powell ldquoAn Iterative Method for Finding Stationary Values of a Functionof Several Variablesrdquo Computer Journal vol 5 no 2 pp 147-151 1962 [AvailableOnline] httpcomjnloxfordjournalsorgcontent52147fullpdf

[65] M J D Powell ldquoAn Efficient Method for Finding the Minimum of a Function forSeveral Variables Without Calculating Derivativesrdquo Computer Journal vol 7 no 2pp 155-162 1964

[66] K V Price R M Storn J A Lampinen ldquoDifferential Evolution A Practical Ap-proach to Global Optimizationrdquo Springer 2005

[67] W L Price ldquoA Controlled Random Search Procedure for Global Optimisa-tionrdquo Computer journal vol 20 no 4 pp 367-370 1977 [Available Online]httpcomjnloxfordjournalsorgcontent204367fullpdf

[68] A Qing ldquoDynamic Differential Evolution Strategy and Applications in Electromag-netic Inverse Scattering Problemsrdquo IEEE Transactions on Geoscience and remote Sens-ing vol 44 no 1 pp 116-125 2006

[69] S Rahnamyan H R Tizhoosh N M M Salama ldquoA Novel Population InitializationMethod for Accelerating Evolutionary Algorithmsrdquo Computers and Mathematics withApplications vol 53 no 10 pp 1605-1614 2007

[70] S Rahnamyan H R Tizhoosh N M M Salama ldquoOpposition-Based DifferentialEvolution (ODE) with Variable Jumping Raterdquo IEEE Sympousim Foundations Com-putation Intelligence Honolulu HI pp 81-88 2007

[71] S S Rao ldquoEngineering Optimization Theory and Practicerdquo John Wiley amp Sons2009

[72] J Ronkkonen ldquoContinuous Multimodal Global Optimization With DifferentialEvolution-Based Methodsrdquo PhD Thesis Lappeenranta University of Technology 2009

[73] H H Rosenbrock ldquoAn Automatic Method for Finding the Greatest or least Value ofa Functionrdquo Computer Journal vol 3 no 3 pp 175-184 1960 [Available Online]httpcomjnloxfordjournalsorgcontent33175fullpdf

[74] R Salomon ldquoRe-evaluating Genetic Algorithm Performance Under Corodinate Rota-tion of Benchmark Functions A Survey of Some Theoretical and Practical Aspects ofGenetic Algorithmsrdquo BioSystems vol 39 no 3 pp 263-278 1996

45

[75] M A Schumer K Steiglitz ldquoAdaptive Step Size Random Searchrdquo IEEE Transactionson Automatic Control vol 13 no 3 pp 270-276 1968

[76] J D Schaffer R A Caruana L J Eshelman R Das ldquoA Study of Control ParametersAffecting Online Performance of Genetic Algorithms for Function Optimizationrdquo Proc3rd International Conf on Genetic Algorithms George Mason Uni pp 51-60 19889

[77] H P Schwefel ldquoNumerical Optimization for Computer Modelsrdquo John Wiley Sons1981

[78] H P Schwefel ldquoEvolution and Optimum Seekingrdquo John Wiley Sons 1995

[79] D F Shanno ldquoConditioning of Quasi-Newton Methods for Function MinimizationrdquoMathematics of Computation vol 24 no 111 pp 647-656 1970

[80] Z K Silagadze ldquoFinding Two-Dimesnional Peaksrdquo Physics of Particles and NucleiLetters vol 4 no 1 pp 73-80 2007

[81] R Storn K Price ldquoDifferntial Evolution - A Simple and Efficient Adaptive Schemefor Global Optimization over Continuous Spacesrdquo Technical Report no TR-95-012International Computer Science Institute Berkeley CA 1996 [Available Online] httpwww1icsiberkeleyedu~stornTR-95-012pdf

[82] P N Suganthan N Hansen J J Liang K Deb Y-P Chen AAuger S Tiwari ldquoProblem Definitions and Evaluation Criteria for CEC2005 Special Session on Real-Parameter Optimizationrdquo Nanyang Techno-logical University (NTU) Singapore Tech Rep 2005 [Available Online]httpwwwlrifr~hansenTech-Report-May-30-05pdf

[83] K Tang X Yao P N Suganthan C MacNish Y-P Chen C-M ChenZ Yang ldquoBenchmark Functions for the CEC2008 Special Session and Competi-tion on Large Scale Global Optimizationrdquo Tech Rep 2008 [Available Online]httpnicalustceducncec08ssphp

[84] K Tang X Li P N Suganthan Z Yang T Weise ldquoBench-mark Functions for the CEC2010 Special Session and Competition onLarge-Scale Global Optimizationrdquo Tech Rep 2010 [Available Online]httpsci2sugreseamhcocec2010_functionspdf

[85] T L Wayburn J D Seader ldquoHomotopy Continuation Methods for Computer-AidedProcess Designrdquo Computers and Chemical Engineering vol 11 no 1 pp 7-25 1987

[86] D Whitley K Mathias S Rana J Dzubera ldquoEvaluating Evolutionary AlgorithmsrdquoArtificial Intelligence vol 85 pp 245ndash276 1996

[87] P H Winston ldquoArtificial Intelligence 3rd edrdquo Addison-Wesley 1992

[88] X S Yang ldquoTest Problems in Optimizationrdquo Engineering Optimization An Intro-duction with Metaheuristic Applications John Wliey amp Sons 2010 [Available Online]httparxivorgabs10080549

[89] X S Yang ldquoFirefly Algorithm Stochastic Test Functions and Design Optimisa-tionrdquo Intl J Bio-Inspired Computation vol 2 no 2 pp 78-84 [Available Online]httparxivorgabs10080549

46

[90] X Yao Y Liu ldquoFast Evolutionary Programmingrdquo Proc 5th Conf on EvolutionaryProgramming 1996

47

• 1 Introduction
• 2 Characteristics of Test Functions
• • 21 Modality
  • 22 Basins
  • 23 Valleys
  • 24 Separability
  • 25 Dimensionality
  • • 3 Benchmark Test Functions for Global Optimization
    • 4 Conclusions