APPENDIX A

TEST PROBLEMS IN OPTIMIZATION

In order to validate any new optimization algorithm, we have to validate it against standard test functions so as to compare its performance with well-established or existing algorithms. There are many test functions, so there is no standard list or set of test functions one has to follow. However, various test functions do exist, so new algorithms should be tested using at least a subset of functions with diverse properties so as to make sure whether or not the tested algorithm can solve certain type of optimization efficiently.

In this appendix, we will provide a subset of commonly used test functions with simple bounds as constraints, though they are often listed as unconstrained problems in literature. We will list the function form f(x), its search domain, optimal solutions tc* and/or optimal objective value /*. Here, we use x = (xi, ...,xn)T where n is the dimension.

Ackley's function:

1 1 1 _ " f(x) = -20exp\--, - V ^ l - e x p [ - V c o s ( 2 r a i ) + 20 + e, (A.l)

L 5 \ n *^ J In 'f J \ i=l i=l

Engineering Optimization: An Introduction with Metaheuristic Applications. 261 By Xin-She Yang Copyright 2010 John Wiley & Sons, Inc.

262 TEST PROBLEMS IN OPTIMIZATION

where n = 1,2,..., and -32.768 < x < 32.768 for i = 1,2,...,n. This function has the global minimum /* = 0 at x* = (0,0,..., 0).

De Jong's functions: The simplest of De Jong's functions is the so-called sphere function

n

f(x) = J2x2i> ~ 5 1 2 - Xi ~ 5 1 2 ' (A2) i=\

whose global minimum is obviously / = 0 at (0,0, ...,0). This function is unimodal and convex. A related function is the so-called weighted sphere function or hyper-ellipsoid function

n

/ ( ;) = ] ? , -5.12 < Xi < 5.12, (A.3)

which is also convex and unimodal with a global minimum /* = 0 at x , = (0,0,..., 0). Another related test function is the sum of different power function

n

f(x) = Yi\xi\i+1, -l

TEST PROBLEMS IN OPTIMIZATION 263

The global minimum /* = 1 of f(x) occurs at x*(l/y/n,...,l/y/n) within the domain 0 < Xi < 1 for i = 1,2, ...,n.

Griewank's function: 1 n n

/ ( x ) = J^) xi - c o s (^) + !> - 6 0 0 ^ x * ^ 6 0 0 ' (A9) W

whose global minimum is /* = 0 at a:* = (0,0, ...,0). This function is highly multimodal.

Michaelwicz's function: n 2 .

f(x) = - s i n ( x i ) sin(L) i = l

2m

where m = 10, and 0 < ^ < pi for i 1,2,..., n. In 2D case, we have

f(x,y) = -sm(x)sm20(-)-Sm(y)sm20(^),

(A.10)

(A.ll)

where (x, y) [0,5] x [0,5]. This function has a global minimum / 1.8013 at x* = (x*,j/*) = (2.20319,1.57049).

Perm Functions: n n

/(*) = {^'+) [ ' -*]}> w > ) (A12) j = l i= l

which has the global minimum /* = 0 at x* = (1, 2,..., n) in the search domain n < Xi + 0 ) N - ' ] }2' (13) j = l i= l

has the global minimum /* = 0 at (1,1/2,1/3, . . . , 1/n) within the bounds 1 < Xi < 1 for all i = 1,2, ...,n. As /? > 0 becomes smaller, the global minimum becomes almost indistinguishable from their local minima. In fact, in the extreme case = 0, every solution is also a global minimum.

Rastrigin's function:

n f(x) = 10n+ } ^ \xf - 10>8(2

= 1

-5.12 < Xi < 5.12, (A.14)

whose global minimum is /* = 0 at (0,0, ...,0). This function is highly multi-modal.

264 TEST PROBLEMS IN OPTIMIZATION

Rosenbrock's function: n - l

f{x) = \(xi - !)2 + 100(zi+i - x2)2 , (A.15)

whose global minimum /* = 0 occurs at x* = (1,1,. . . , 1) in the domain 5 < Xi < 5 where i = 1,2, ...,n. In the 2D case, it is often written as

/ (x ,y) = ( x - l ) 2 + 100(y-x 2 ) 2 , (A.16)

which is often referred to as the banana function.

Schwefel's function: n

f(x) = -Y^Xism(^y/\xi\\, -500 < Xi < 500, (A.17) i=\

whose global minimum /* 418.9829 occurs at Xj = 420.9687 where % = 1,2, ...,n.

Six-hump camel back function:

fix, y) = (4 - 2.1x2 + l-xA)x2 + xy + 4(y2 - l)y2, (A.18)

where 3 < x < 3 and 2 < y < 2. This function has two global minima /* -1.0316 at (x*,y*) = (0.0898,-0.7126) and (-0.0898,0.7126).

Shubert's function: n n

/ ( ) = [ 5 ^ t c o s ( t + (i + l)a:)] [ 5 ^ i c o s ( i + ( i + l ) y ) , (A.19) i= l i= l

which has 18 global minima / w 186.7309 for n = 5 in the search domain - 1 0 < x , y < 10.

Xin-She Yang's functions: n n

fix)= ( ^ I x i l j e x p f - ^ s i n i x 2 ) ] , (A.20)

which has the global minimum / = 0 at x* = (0,0,...,0) in the domain 2 < Xi < 2pi where i = 1,2, ...,n. This function is not smooth, and its derivatives are not well defined at the optimum (0,0,..., 0).

A related function is n n

/(a;) = - ( ^ | x i | ) e x p ( - ^ x 2 ) , - 1 0 < x 8 < 1 0 , (A.21) i= l i= l

TEST PROBLEMS IN OPTIMIZATION 265

which has multiple global minima. For example, for n = 2, we have 4 equal minima /* = - l / \ / e -0.6065 at (1/2,1/2), (1 /2 , -1 /2 ) , ( -1 /2 ,1 /2) and ( - 1 / 2 , - 1 / 2 ) .

Yang also designed a standing-wave function with a defect

n f(x)= r e -Er= i ( x < / / 3 ) 2 m -2e -Sr= 1 ^ l . J J c o s 2 ^ , m = 5, (A.22)

*=i

which has many local minima and the unique global minimum /* = 1 at x* = (0,0,..., 0) for = 15 within the domain 20 < Xi < 20 for i = 1,2,..., n. He also proposed another multimodal function

fix) = { [ sin2(x,)] - exp( - *?)} " i n 2 ^ / ^ ] , (.23) i= l i= l i\

whose global minimum /* = 1 occurs at x* = (0,0, ...,0) in the domain 10 < Xi < 10 where i 1,2,..., n. In the 2D case, its landscape looks like a wonderful candlestick.

Most test functions are deterministic. Yang designed a test function with stochastic components

f{x, y) = - - ^ - ^ + * - * ) ' ] - ^-[(*-*)2+(^)2], (.24) 3 = 1 i= l

where a, > 0 are scaling parameters, which can often be taken as = = 1. Here ey are random variables and can be drawn from a uniform distribution ij ~ Unif[0,l]. The domain is 0 < x, y < K and K = 10. This function has K2 local valleys at grid locations and the fixed global minimum at x* = (, ). It is worth pointing that the minimum /m;n is random, rather than a fixed value; it may vary from (K2 + 5) to 5, depending a and as well as the random numbers drawn.

Furthermore, he also designed a stochastic function

71 I 1 f{x) = V i \Xi - - , - 5 < Xi < 5, (A.25)

i= l

where ti (z = 1,2,..., n) are random variables which are uniformly distributed in [0,1]. That is, e^ ~Unif[0,1]. This function has the unique minimum /* = 0 at x* (1,1/2,..., 1/n) which is also singular.

Zakharov's functions:

z = l i=l i=l

266 TEST PROBLEMS IN OPTIMIZATION

whose global minimum / = 0 occurs at a; = (0,0, . . . ,0). Obviously, we can generalize this function as

K

/(*)=5>? + E./f, (A.27)

where K = 1,2, ...,20 and

i = l fc=l

. (.28) 1 i=\

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