research article physics-inspired optimization algorithms...

Hindawi Publishing CorporationJournal of OptimizationVolume 2013 Article ID 438152 16 pageshttpdxdoiorg1011552013438152

Research ArticlePhysics-Inspired Optimization Algorithms A Survey

Anupam Biswas K K Mishra Shailesh Tiwari and A K Misra

Department of Computer Science amp Engineering Motilal Nehru National Institute of Technology Allahabad Allahabad 211004 India

Correspondence should be addressed to K K Mishra mishrakrishngmailcom

Received 7 February 2013 Revised 22 May 2013 Accepted 24 May 2013

Academic Editor Qingsong Xu

Copyright copy 2013 Anupam Biswas et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Natural phenomenon can be used to solve complex optimization problems with its excellent facts functions and phenomenonIn this paper a survey on physics-based algorithm is done to show how these inspirations led to the solution of well-knownoptimization problem The survey is focused on inspirations that are originated from physics their formulation into solutionsand their evolution with time Comparative studies of these noble algorithms along with their variety of applications have beendone throughout this paper

1 Introduction

LeonidKantorovich introduced linear programming for opti-mizing production in plywood industry in 1939 and probablyit was the first time the term optimization of a process wasused though Fermat and Lagrange used calculus for findingoptima andNewton andGauss proposedmethods formovingtowards an optimum Every technological process has toachieve optimality in terms of time and complexity and thisled the researchers to design and obtain best possible orbetter solutions In previous studies several mathematicalsolutions were provided by various researchers such as LP[1] NLP [2] to solve optimization problems The complex-ity of the proposed mathematical solutions is very highwhich requires enormous amount of computational workTherefore alternative solutions with lower complexity areappreciated With this quest nature-inspired solutions aredeveloped such as GA [3] PSO [4] SA [5] and HS [6]These nature-inspired metaheuristic solutions became verypopular as the algorithms provided are much better in termsof efficiency and complexity than mathematical solutionsGenerally these solutions are based on biological physicaland chemical phenomenon of nature

In this paper the algorithms inspired by the phenomenonof physics are reviewed surveyed and documented Thispaper mainly focuses on the following issues

(i) most inspirational facts and phenomena(ii) their formulation into a solution

(iii) parameters considered for this formulation(iv) effectiveness of these parameters(v) variation with time in inspiration(vi) other biological influences(vii) convergence exploration and exploitation(viii) Various applications

The rest of the paper is organized as follows Section 2overviews the history of physics-inspired algorithms andalso the description of few major algorithms In Section 3 acorrelative study of these major algorithms is done on thebasis of their inspirational theory and formulation methodVarious parameters used in these algorithms along with theirvariants and respective applications are also discussed in thissection In Section 4 finally the overall study is concluded

2 Historical Study

Both simplicity and efficiency attract researchers towardsnatural phenomenon resulting in some popular algorithmssuch as GA [3] based on Darwinrsquos principle of survival ofthe fittest SA [5] in 1983 based on the annealing process ofmetal PSO [4] in 1995 based on the behavior of fishes andbirds swarms andHS [6] in 2001 based on the way amusicianadjusts instruments to obtain good harmony Richard Feyn-manrsquos proposal of quantum computing system [7 8] inspired

2 Journal of Optimization

QGA QEA EM QPSO

BB-BC

HQGA

RQGA

QSE

vQEA

IQEA

CFO

BQEA

QICA

HO

VM-APO

EAPO

BGSA

APO

GSA

UBB-CBC

PSOGSA

MOGSA

CQACO

CSS

GIO

GbSA

ECFO

IGOA

QBSO

1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014Timeline

Figure 1 Evolution of physics-inspired optimization algorithms

by quantum mechanics in 1982 paved way for physics-inspired optimization algorithms With this the concept ofquantum computing was developed and in 1995 Narayananand Moore [9] proposed Quantum-Inspired Genetic Algo-rithm (QGA) This is the beginning of physics-inspired opti-mization algorithms After half a decade later in 2002 Hanand Kim proposed Quantum-Inspired Evolutionary Algo-rithm (QEA) In 2004 Quantum-Inspired Particle SwarmOptimization (QPSO) was proposed by Sun et al [10] andin 2007 another swarm-basedQuantum Swarm EvolutionaryAlgorithm (QSE) was proposed by Wang et al [11] Apartfrom the quantummechanics other principles and theoremsof physics also begun to draw the attention of researchersIn 2003 Birbil and Fang [12] proposed Electromagnetism-like (EM) mechanism based on the superposition principleof electromagnetism Big Bang-Big Crunch (BB-BC) [13]based on hypothetical theorem of creation and destructionof the universe was proposed in 2005 Based on Newtonrsquosgravitational law and laws of motion algorithms emergedsuch as CFO [14] by Formato in 2007 GSA by Rashediet al [15] APO by Xie et al [16] in 2009 and GIO byFlores et al [17] in 2011 Hysteretic Optimization (HO) [18]based on demagnetization process was proposed in 2008In 2010 Kaveh and Talatahari proposed CSS [19] based onelectrostatic theorems such as Coulombrsquos law Gaussrsquos law andsuperposition principle from electrostatics andNewtonrsquos lawsof motion In 2011 Shah-Hosseini proposed Spiral Galaxy-Based SearchAlgorithm (GbSA) [20] Jiao et al [21] proposedQICA in 2008 based on quantum theory and immune systemLi et al [22] proposed CQACO based on quantum theoryand ant colony in 2010 Most recently in 2012 Zhang etal [23] proposed IGOA based on gravitational law andimmune system and Jinlong and Gao [24] proposed QBSObased on quantum theory and bacterial forging These majoralgorithms along with their modified improved and hybridversions alongwith the year of proposal are shown in Figure 1We have categorized these algorithms with their variants andtheir notion of inspiration as follows

(A) Newtonrsquos gravitational law(i) Pure physics

(1) CFO(a) Variant (pure physics)(1) ECFO

(2) APO(a) Variant (pure physics)(1) EAPO(2) VM-APO

(3) GSA(a) Variant (pure physics)(1) BGSA(2)MOGSA

(b) Variant (Semiphysics)(1) PSOGSA

(4) GIO(ii) Semiphysics(1) IGOA

(B) Quantum mechanics(i) Pure physics(1) QGA(a) Variant (pure physics)(1) RQGA(2) QGO

(b) Variant (semiphysics)(1)HQGA

(2) QEA(a) Variant (pure physics)(1) BQEA(2) vQEA(3) IQEA

(ii) Semiphysics(1) QPSO(2) QSE(3) QICA(4) CQACO(5) QBSO

(C) Universe theory(i) Pure physics(1) BB-BC(a) Variant (pure physics)(1) UBB-CBC

(2) GbSA

Journal of Optimization 3

(D) Electromagnetism(i) Pure physics(1) EM

(E) Glass demagnetization(i) Pure physics(1)HO

(F) Electrostatics(i) Pure physics(1) CSS

3 Algorithms

31 Newtonrsquos-Gravitation-Law-Based Algorithms

311 CFO CFO [14] is inspired by the theory of particlekinematics in gravitational field Newtonrsquos universal law ofgravitation implies that larger particles will have more attrac-tion power as compared to smaller particles Hence smallerones will be attracted towards the larger ones As a result allsmaller particles will be attracted towards the largest particleThis largest particle can be resembled as global optimumsolution in case of optimization To mimic this concept inCFO a set of solutions is considered as probes on the solutionspace Each probe will experience gravitational attraction dueto the other Vector acceleration experienced by probe 119901withrespect to other probes at iteration t is given by the equationbelow

119886119901

119905 = 119866

119873

sum119896=1119896 = 119901

119880(119865119896

119905minus 119865119901

119905 ) sdot (119865119896

119905minus 119865119901

119905 )120572

times(119877119896119905minus 119877119901

119905 )

10038171003817100381710038171003817119877119896119905 minus 119877

119901

119905

10038171003817100381710038171003817

120573 (1)

Here 119866 is CFOrsquos gravitational constant 119877119901119905 and 119865119901

119905 are theposition of a probe 119901 and objective function value at thatposition respectively at iteration 119905 119877119896

119905and 119865119896

119905are the posi-

tion of all other probe and objective function value at thatposition respectively at iteration 119905 119880(sdot) is the unit stepfunction The CFO exponents 120572 and 120573 by contrast have noanalogues in nature but these exponents provide flexibility tothe algorithmThese parameters have drastic effect on overallexploration and convergence of the algorithmThe algorithmdoes not have any apparent mechanism for exploitation

In this equation 119880(119865119896119905minus 119865119901

119905 ) sdot (119865119896

119905minus 119865119901

119905 )120572

defines CFOrsquosmass which is analogous to real objects mass in space

The 119886119901119905 causes the probe 119901 to move from position 119877119901119905to 119877119901119905+1

and the new location is obtained by the followingequation

119877119901

119905+1= 119877119901

119905 +1

2119886119901

119905 Δ1199052 (2)

Here Δ119905 is the time interval between iterations RecentlyDing et al proposes an extended version of CFO namelyECFO [25] Applications of this algorithm are neural network[26] and antenna applications [27 28]

312 APO APO [16] is based on the concept of artificialphysics or physicomimetics [29] whichwas applied to robots

Analogous toNewtonrsquos gravitation law a new kind of force lawis defined as follows

119865 = 11986611989811198982

119903119901 (3)

where 119865 is the force exerted between particles1198981 and1198982 in ahypothetical universe119866 is gravitational constant and 119903 is thedistance between particles 1198981 and 1198982 Unlike real universethe value of 119901 is not always equal to 2 instead it varies fromminus5 to +5

Mass is defined in APO as follows

119898119894 = 119890(119891(119909best)minus119891(119909119894))(119891(119909worst)minus119891(119909best)) (4)

Considering value of 119901 = minus1 in (3) the force in APO isdefined as follows

119865119896

119894119895=

119866119898119894119898119895 (119909119896

119895minus 119909119896119894) if119891 (119883119895) lt 119891 (119883119894)

minus119866119898119894119898119895 (119909119896

119895minus 119909119896119894) if119891 (119883119895) ge 119891 (119883119894)

(5)

where 119865119896119894119895is the 119896th component of force exerted on par-

ticle 119894 by particle 119895 119909119896119894and 119909119896

119895are the 119896th dimension of

particles 119894 and 119895 respectively The 119896th component of thetotal force 119865119896

119894exerted on particle 119894 by all other particles is

given by the following

119865119896

119894=

119873

sum119895=1

119865119896

119894119895forall119894 = best (6)

Velocity and positions of particles are updated with followingequation

V119896119894(119905 + 1) = 119908V119896

119894(119905) +

120582119865119896

119894

119898119894

119909119896

119894(119905 + 1) = 119909

119896

119894(119905) + V119896

119894(119905 + 1)

(7)

where 120582 is uniformly distributed random variable in [0 1] 119908is user-defined weight 0 lt 119908 lt 1

Main exploitation and convergence component of APOalgorithm is the computation of force exerted on each particleby others Overall exploration of algorithm is controlledby the weight parameter 119908 The parameter 120582 is actually forputting limitation to convergence But due to randomnessit also serves for exploration To overcome the lack ofconvergence component an extended version of APO isproposed in [30] where individual particlersquos best position istracked in iteration and utilized in velocity updating A vectormodel of APO is defined in [31]

313 GSA GSA [15] is inspired by Newtonrsquos law of universalgravitation and law ofmotion In addition to this another factof physics is also considered according to which the actualvalue of gravitational constant119866 depends on the actual age ofthe universe So 119866 at time 119905 can be expressed as follows

119866 (119905) = 119866 (1199050) times (1199050

119905)120573

120573 lt 1 (8)


where 119866(1199050) is the value of the gravitational constant atthe first cosmic quantum-interval of time 1199050 120573 is a time-dependent exponent

In GSA the solution space is considered as an imaginaryuniverse Every point in solution space is considered asan agent having mass To compute mass of any agent 119894 aparameter119898119894(119905) is computed The parameter119898119894(119905) and mass119872119894(119905) of agent 119894 are computed as follows

119898119894 (119905) =fit119894 (119905) minus worst (119905)best (119905) minus worst (119905)

119872119894 (119905) =119898119894 (119905)

sum119873

119895=1119898119895 (119905)

(9)

where fit119894(119905) is the fitness value of the agent 119894 at time 119905Force exerted on each considered agent is computed as

follows

119865119889

119894119895= 119866 (119905)

119872119901119894 (119905) times 119872119886119895 (119905)

119877119894119895 (119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

119877119894119895 =10038171003817100381710038171003817119883119894 (119905) 119883119895(119905)

100381710038171003817100381710038172

119865119889

119894(119905) =

119873

sum119895=1119895 = 119894

rand119895119865119889

119894119895(119905)

(10)

where 119865119889119894119895is the force acting on mass119872119894 and mass119872119895 at time

119905119872119886119895 is the active gravitational mass related to agent 119895 119872119901119894is the passive gravitational mass related to agent 119894 119866(119905) isgravitational constant at time 119905 120576 is a small constant 119877119894119895(119905)is the Euclidian distance between two agents 119894 and 119895 119865119889

119894(119905) is

the total force that acts on agent 119894 in a dimension 119889 at time 119905and rand119895 is a random number in the interval [0 1]

Acceleration of any agent 119894 at time 119905 in direction 119889 iscomputed with equation given below

119886119889

119894(119905) =

119865119889119894

119872119894 (119905) (11)

Thenext position of each agent and at which velocity theywillmove is calculated as follows

V119889119894(119905 + 1) = rand119894 times V119889

119894(119905) + 119886

119889

119894(119905)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1)

(12)

The concept of variable gravitational constant 119866 providesa good mechanism for convergence to the algorithm Asin subsequent iterations the value of 119866 gradually increasesattraction force experienced by each agent also increasesThus agents converge towards the better agents with incre-mental attraction However the effect of attraction force iscontrolled by a random parameter This random control offorce ensures exploitation as well as exploration Anotherrandom parameter used in velocity updating also impliesexploration of search space

In [32] binary version of GSA is proposed multiobjectiveGSA [33] is proposed by Mirjalili and Hashim and a hybridof PSO and GSA is proposed in [34]

Applications of GSA algorithm are in power system[35ndash42] economic dispatch problem [43 44] Wessingerrsquosequation [45] fuzzy system [46 47] forecasting of futureoil demand [48] slope stability analysis [49] clustering [50ndash52] prototype classification [53] feature selection [54] webservices [55] PID controller [56] antenna application [47]and so forth

314 IGOA IGOA [23] algorithm is an improved versionof GSA [15] The gravitation-law-based algorithm GSA caneasily fall into local optimum solution and convergence rateis also comparatively slow [57] To overcome these problemsIGOA introduces new operators which are inspired frombiological immune system (BIS) [58] In BISmainly two kindsof activities take place activities of antigens and activities ofantibodyAn antigen can only be negotiatedwith correspond-ing right antibody which comes from mother during birthBut for an unknown antigen BIS also can act accordinglyby learning That means BIS has immune memory andantibody diversity IGOA mimics this mechanism to avoidfalling into local optimum In this case local optimum issimilar to the unknown antigen in BIS In IGOA vaccinationand memory antibody replacement is used to improve theconvergence speed and antibody diversity mechanism tocatch the diversity of solution space along with GSA IGOA isa newly proposed algorithm and not yet applied in any real-life application

315 GIO GIO [17] algorithm is similar to GSA [15] andCSS[19] where each point in the search space is assigned massand charges respectively Although perspective of assigningmasses or charges to each point is similar the way ofassignment and notion is different CSS is inspired fromelectrostatic dynamics law whereas GSA is inspired fromNewtonrsquos gravitational laws and laws of motion GIO is alsoinspired from Newtonrsquos law but unlike GSA this algorithmkeeps hypothetical gravitational constant119866 as constant Forceexerted between two bodies is computed as follows

119865119894119895 =119872(119891 (119861119894)) sdot 119872 (119891 (119861119895))

10038161003816100381610038161003816119861119894 minus 119861119895

10038161003816100381610038161003816

2119861119894119895 (13)

where 119861119894 is the position of the 119894th body and 119861119895 is that of119895th body 119865119894119895 is exerting force on the mass 119861119894 |119861119894 minus 119861119895| isthe Euclidean distance and 119861119894119895 is the unit vector betweenbodies 119861119894 and 119861119895 119891(119861119894) is the fitness of body 119861119894 119872(119891(119861119894))is corresponding mass of body 119861119894 and is computed as follows

119872(119891 (119861119894)) = (119891 (119861119894) minus min 119891 (119861)

max119891 (119861) minusmin119891 (119861)

times (1 minusmapMin) +mapMin)2

(14)

wheremin119891(119861) is theminimumfitness value of the positionsof the bodies so far max119891(119861) is the maximum fitness valueof the positions of the bodies so far mapMin is a constantused to limit the fitness value 119891(119861119894) to a mass in the interval


[mapMin 1) As each body are interacts with other bodies soresultant force acting on body 119861119894 is computed as follows

119865119894 =

119899

sum119895=1

119872(119891 (119861119894)) sdot 119872 (119891 (119861best119895))

10038161003816100381610038161003816119861119894 minus 119861

best119895

10038161003816100381610038161003816

2119861119894119861

best119895 (15)

Velocity with which a body will move to its new position iscomputed as follows

119881119905+1 = 120594 (119881119905 + 119877 sdot 119862 sdot 119861119896) (16)

where 119881119905 is the current velocity of 119861119894 119877 is a random realnumber generated in the range of [0 1) 119862 is the gravitationalinteraction coefficient 119861119896 is the displacement of body 119861119894 andis computed with (17) and 120594 is the inertia constraint and iscomputed with (18)

119861119896 = radic119872(119891 (119861119894))

10038161003816100381610038161198651198941003816100381610038161003816

119865119894 (17)

120594 =2119896

10038161003816100381610038161003816100381610038162 minus 120601 minus radic1206012 minus 4120601

1003816100381610038161003816100381610038161003816

(18)

In (18) 119896 is an arbitrary value in the range (0 1] 120601 = 1198621+1198622 gt4 where 1198621 and 1198622 are the cognitive and the gravitationalinteraction constants respectively

New position of a body is obtained by adding the com-puted velocity corresponding to it Formula given above forcomputing velocity is for unimodal optimization problemswhich is further modified for multimodal optimization prob-lems as follows

119881119905+1 = 120594 (119881119905 + 1198621 sdot 1198771 sdot (119861bestminus 119861) + 1198622 sdot 1198772 sdot 119861119896) (19)

Here 1198771 and 1198772 are real random numbers in the range [0 1)New position for next iteration is obtained by adding the

updated velocity with current body as follows

119861119905+1 = 119861119905 + 119881119905+1 (20)

Certain precaution has been taken during resultant forcecomputation in order to avoid numerical errors The forcebetweenmasses119872(119861119894) and119872(119861

best119895) is computed only if |119861119894minus

119861119895| ge 1times10minus5 In order to avoid division by 0 119861119896 is computed

only if for a body 119861119894 resultant force 119865119894 gt 0The concept of inertia constant 120594 is similar to the con-

cept of constriction parameter in constricted PSO [59ndash61]Exploration of a body in GIO is controlled by this parameterExploration exploitation and convergence are ensured bycomputation of mass and resultant forceThe inertia constant120594 also helps in convergence Though Flores et al [17] showsGIOrsquos superiority over PSO inmultimodal problems but it hasnot been yet applied in any real-life application

32 Quantum-Mechanics-Based Algorithms

321 QGA According to quantum mechanics electrons aremoving around the nucleus in an arc path known as orbitsDepending on the angular momentum and energy level

electrons are located in different orbits An electron in lowerlevel orbit can jump to higher level orbit by absorbing certainamount of energy similarly higher level electron can jumpto lower energy level by releasing certain amount of energyThis kind of jumping is considered as discrete There is nointermediate state in between two energy levels The positionwhere an electron lies on the orbit is unpredictable it may lieat any position in orbit at a particular time Unpredictabilityof electronrsquos position is also referred as superposition ofelectron

In classical computing a bit is represented either by 0 or1 but in quantum computing this is termed as qubit State of aqubit can be 0 or 1 or both at the same time in superpositionstate This superposition of qubit mimics the superpositionof electrons or particles State of qubit at any particular timeis defined in terms of probabilistic amplitudes The positionof an electron is described in terms of qubits by a vectorcalled quantum state vector A quantum state vector can bedescribed with the equation given below

|Ψ⟩ = 120572| 0⟩ + 1205731003816100381610038161003816 1⟩ (21)

where 120572 and 120573 are complex numbers that specify the proba-bility amplitudes of obtaining the qubit in ldquo0rdquo state and in ldquo1rdquostate respectively In this case the value of 120572 and 120573 alwayssatisfies the equation |120572|2 + |120573|2 = 1 For 119899 positions ofelectrons states can be described by 119899 state vectors These 119899positions of an electron can be known simultaneously

QGA [9] utilized the concept of parallel universe in GA[3] to mimic quantum computing According to this paralleluniverse interpretation each universe contains its own ver-sion of population All populations follow the same rules butone universe can interfere in population of other universeThis interference occurs as in the form of a different kindof crossover called interference crossover which providesgood exploration capability to the algorithm In QGA all thesolutions are encoded using superposition and all of thesesolutions may not be valid which creates problems duringimplementation of crossover Udrescu et al propose RQGA[62] which provides a mechanism to overcome this problemHybrid versions [63] merge QGA with permutation-basedGA and [64] merge QGA with real-valued GA Malossiniand Calarco propose QGOA [65] very similar to QGAwith special quantum-based selection and fitness evaluationmethods

Many applications have been developed in recent years onthe basis of this algorithm such as structural aligning [66]clustering [67 68] TSP [69] combinatorial optimizationproblem [70] web information retrieval [71] computationalgrid [72] software testing [73] dynamic economic dispatch[74] area optimization [75] operation prediction [76] com-puter networking [77 78] PID controller [79] multivariateproblem [80] course timetabling [81] minimal redact [82]image applications [83ndash86] smart antenna [87] hardware[88] fuzzy system [89 90] neural network [91] and robotapplication [92]

322 QEA Quantum bit and superposition of states arethe main basis of this algorithm QEA [93] is originally


inspired by quantum computing which itself is inspired bythe quantummechanics In QEA the state of a qubit or Q-bitis represented as pair of numbers (120572 120573) in a column matrix[120572120573 ] where |120572|2 + |120573|2 = 1 and |120572|2 gives the probability that

the Q-bit will be found in the ldquo0rdquo state and |120573|2 gives theprobability that the Q-bit will be found in the ldquo1rdquo state

A Q-bit individual which is a string of Q-bits is defined asfollows

[12057211205731

10038161003816100381610038161003816100381610038161003816

12057221205732

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120572119898120573119898

] (22)

where |120572119894|2+ |120573119894|

2= 1 119894 = 1 2 119898 With this Q-bit

representation a population set is formulated and opera-tions are performed on that population Zhang and Gaofurther improved this algorithm as IQEA [94] by intro-ducing probability amplitude ratio 120574120572 = |120573||120572| if 120572 = 0

and 120574120573 = |120572||120573| if 120573 = 0 to define relative relationshipbetween 120572 and 120573 As quantum rotation gate is unable tocover the entire search space since it outputs discrete valuesa mechanism for calculating rotation angle of quantum rota-tion gate is defined Platel et al propose versatile QEA [95]with introducing new concept of hitchhiking phenomenoninto QEA with little bit elitism in updating parameters andP Li and S Li propose Bloch QEA [96] based on Blochcoordinates depicted by qubits Here 120572 and 120573 are defined ascos(1205792) and 119890119894120593 sin(1205792) respectively This 120579 and 120593 definebloch points

Applications of QEA-related algorithms are combinato-rial optimization [97 98] image segmentation [99] Knap-sack Problems [100ndash102] resource optimization [103 104]numerical optimization [105 106] extrusion [107] unit com-mitment problem [108 109] power system [110 111] signaling[112] face identification [113 114] financial data analysis [115]Option pricing model calibration [116 117] stock marketprediction [118] and so forth

323 QSE QSE [11] takes the concepts from both QEA[93] and PSO [4] Similar to PSOrsquos swarm intelligent con-cept quantum swarms are represented using Q-bits UnlikeQEA representation of Q-bit in QSE changes probabilisticparameters 120572 and 120573 are replaced with angular parame-ters sin 120579 and cos 120579 here 120579 is quantum angle Q-bit [120579] isrepresented as [ sin 120579cos 120579 ] where | sin 120579|

2+ | cos 120579|2 = 1 For119898Q-

bits this can be represented as [ sin 1205791 sin 1205791 sdotsdotsdot sin 1205791cos 1205791 cos 1205791 sdotsdotsdot cos 1205791 ] Each bit

position 119909119905119894119895of each individual at time 119905 is determined with

the following

119909119905

119894119895=

1 if random [0 1] gt10038161003816100381610038161003816cos 120579119894119895

10038161003816100381610038161003816

2

0 otherwise(23)

Velocity is updated as in PSO Another quantum-swarm-based PSO called QPSO was proposed by Sun et al [10]Unlike QSE state of particle is not determined by theprobabilistic angular parameters Here state of particle isdetermined by a wave function Ψ(119909) as follows

Ψ (119909) =1

radic119871119890minus(119888minus119909119871)

(24)

Here 119888 and 119909 are the center or current best and current loca-tion vector 119871 is called creativity or imagination parameter ofparticle Location vector is defined as

119909 (119905) = 119888 plusmn119871

2ln( 1

119877) (25)

Here 119877 is a random number in range [0 1] The creativityparameter 119871 is updated as follows

119871 (119905 + 1) = 2 times 120572 times |119888 minus 119909 (119905)| (26)

Here 120572 is the creative coefficient and acts as main ingredientfor convergence towards the optima Huang et al [119] haveimproved this later on by considering global best instead ofcurrent best

Applications of these algorithms are flow shop scheduling[120] unit commitment problem [121 122] neural network[123] power system [124ndash126] vehicle routing problem [127ndash129] engineering design [130 131] mining association rules[132] and so forth

324 QICA Basic concept ofQICA [21] is Artificial ImmuneSystemrsquos clonal selection which is hybridized with theframework of quantum computing Basic quantum repre-sentational aspect is similar to QEA [93] QICA introducessome new operators to deal with premature convergenceand diverse exploration The clonal operator Θ is defined asfollows

Θ (119876 (119905)) = [Θ (1199021)Θ(1199022) sdot sdot sdot Θ (119902119899)]119879 (27)

where 119876(119905) is quantum population and Θ(119902119894) = 119868119894119902119894 119868119894 is theidentity matrix of dimensionality 119863119894 which is given by thefollowing

119863119894 = lceil119873119888 times119860 (119902119894)

sum119899

119894=1119860 (119902119894)

rceil (28)

Here 119860(sdot) is function for adaptive self-adjustment and119873119888 isa given value relating to the clone scale After cloning theseare added to population

The immune genetic operator consists of two mainparts that is quantum mutation and recombination Beforeperforming quantummutation population is guided towardsthe best one by using following equation

[

[

120572119906

119894

120573119906

119894

]

]

= 119880 (120579119894) times[

[

120572119901

119894

120573119901

119894

]

]

(29)

where120572119906119894and 120573119906

119894are updated values120572119901

119894and 120573119901

119894are previous

values of probabilistic coefficients 119880(120579119894) = [119903119897 sin 120579119894 minus sin 120579119894cos 120579119894 cos 120579119894 ]

is quantum rotation gate and 120579119894 is defined as follows

120579119894 = 119896 times 119891 (120572119894 120573119894) (30)

where 119896 is a coefficient which determines the speed ofconvergence and the function119891(120572119894 120573119894) determines the search


directionThis updated population ismutated using quantumNOT gate as

119876119906(119905) = radic1 minus |119876119906 (119905)|

2 (31)

Quantum recombination is similar to interference crossoverin QGA [9] Finally the clonal selection operator selects thebest one from the population observing the mutated oneand original population Clonal operator of QICA increasesexplorative power drastically in contrast to QEA

325 CQACO CQACO [22] merges quantum computingand ACO [133] Antrsquos positions are represented by quantumbitsThis algorithm also represents qubits similar toQEA [93]and QICA [21] and uses the concept of quantum rotationgate as in QICA Similar to CQACO Wang et al [134]proposed quantum ant colony optimization Another variantis proposed by You et al [135] in 2010 Quantum concept withACO provides good exploitation and exploration capabilityto these algorithms Applications of these algorithms are faultdiagnosis [136] robot application [137] and so forth

326 QBSO QBSO [24] is the newest among all thequantum-based algorithms This algorithm is semiphysics-inspired as it incorporates concepts of both bacterial forgingand quantum theory In other words QBSO is an improvedversion of BFO [138] As BFO is unable to solve discrete prob-lemsQBSOdeals with this problemby using quantum theoryto adapt the process of the BFO to accelerate the convergencerate BFO consists of chemotaxis swarming reproductionelimination and dispersal processes whereas QBSO consistsmainly of three of them chemotaxis reproduction andelimination dispersal In QBSO also the qubit is defined as in(21) Quantumbacterium of S bacteria is represented in termsof the three processes that is bit position chemotactic stepand reproduction loop

The 119894th quantum bacteriumrsquos quantum 119895th bit at the 119898thchemotactic step of the 119899th reproduction loop in the 119901thelimination dispersal event is updated as follows

Ψ119894119895 (119898 + 1 119899 119901)

=

radic1 minus (Ψ119894119895 (119898 119899 119901))2

if (120579119894+1119894119895= 0 and 120588 (119898 + 1 119899 119901) lt 1198881)

abs(Ψ119894119895 (119898 119899 119901) times cos 120579119894+1

119894119895

minusradic1 minus (Ψ119894119895 (119898 119899 119901))2

times sin 120579119894+1119894119895

)

otherwise(32)

where 120579119905+1119894119895

is the quantum rotation angle which is calculatedthrough (33) 119905 is the iteration number of the algorithm 120588119894119895 isuniform random number in range [0 1] and 1198881 is mutationprobability which is a constant in the range [0 1]

After updating quantum bacterium the correspondingbit position in the population 119909119894119895 is updated with (34) where120574119894119895 is uniform random number between 0 and 1

120579119894+1

119894119895= 1198901 (119887119895 (119898 119899 119901) minus 119909119894119895 (119898 119899 119901)) (33)

Here 1198901 is attracting effect factor and 119887119895 is the 119895th bit positionof global optimal bit

119909119894119895 (119898 + 1 119899 119901)

=

1 if 120574119894119895 (119898 + 1 119899 119901) gt (Ψ119894119895(119898 + 1 119899 119901))2

0 if 120574119894119895 (119898 + 1 119899 119901) le (Ψ119894119895(119898 + 1 119899 119901))2

(34)

Fitness value of each point solution in population is repre-sented as the health of that particular bacterium

33 Universe-Theory-Based Algorithms

331 BB-BC BB-BC [13] algorithm is inspired mainly fromthe expansion phenomenon of Big Bang and shrinking phe-nomenon of Big Crunch The Big Bang is usually consideredto be a theory of the birth of the universe According to thistheory all space timematter and energy in the universe wereonce squeezed into an infinitesimally small volume and ahuge explosion was carried out resulting in the creation ofour universe From then onwards the universe is expandingIt is believed that this expansion of the universe is due to BigBang However many scientists believe that this expansionwill not continue forever and all matters would collapse intothe biggest black hole pulling everything within it which isreferred as Big Crunch

BB-BC algorithm has two phases namely Big Bangphase and Big Crunch phase During Big Bang phase newpopulation is generated with respect to center of massDuring Big Crunch phase the center of mass is computedwhich resembles black hole (gravitational attraction) BigBang phase ensures exploration of solution space Big Crunchphase fullfills necessary exploitation as well as convergence

BB-BC algorithm suffers botching all candidates into alocal optimum If a candidate with best fitness value con-verges to an optima at the very beginning of the algorithmthen all remaining candidates follow that best solution andtrapped into local optima This happens because the initialpopulation is not uniformly distributed in the solution spaceSo this algorithm provides a methodology to obtain uniforminitial population in BB-BC Initially that is at level 1 twocandidates 1198621 and 1198622 are considered at level 2 1198621 and 1198622are subdivided into 1198623 and 1198624 at level 3 1198621 and 1198622 are againdivided into 1198624 1198625 1198626 1198627 1198628 1198629 and 11986210 and so on Thiskind of division continues until we get the required numbersof candidates for initial population In this way at 119899th level1198621and 1198622 are subdivided into 2119899 minus 2 candidates and include inpopulation In addition to this in Big Crunch phase chaoticmap is introduced which improves convergence speed ofalgorithm In this Chaotic Big Crunch phase next positionof each candidate is updated as follows

119909new119894

= 119909119888 plusmn120572 (119905) (119909max minus 119909min)

119905 (35)


where 120572119905+1 = 119888119891(120572(119905)) 0 lt 120572(119905) lt 1 here 119888119891(119909) is a chaoticmap or function BB-BC with uniform population is calledUBB-BC and with chaotic map is called BB-CBC If both areused then it is called UBB-CBC

Applications of this algorithm are fuzzy system [139ndash141] target tracking [142 143] smart home [144] coursetimetabling [145] and so forth

332 GbSA GbSA [20] is inspired by spiral arm of spiralgalaxies to search its surrounding This spiral movementrecovers from botching into local optima Solutions areadjusted with this spiral movement during local search aswell This algorithm has two components

(1) SpiralChaoticMove(2) LocalSearch

SpiralChaoticMove actually mimics the spiral arm nature ofgalaxies It searches around the current solution by spiralmovement This kind of movement uses some chaotic vari-ables around the current best solution Chaotic variables aregenerated with formula 119909119899+1 = 120582119909119899(1 minus 119909119899) Here 120582 = 4

and 1199090 = 019 In this way if it obtains a better solutionthan the current solution it immediately updates and goesfor LocalSearch to obtain more suitable solution aroundthe newly obtained solution GbSA is applied to PrincipleComponent Analysis (PCA) LocalSearch ensures exploita-tion of search space and SpiralChaoticMove provides goodexploration mechanism of search space ensuring reachabilityof algorithm towards the global optimum solution

34 Electromagnetism-Based Algorithms

341 Electromagnetism-Like EM EM[12] algorithm is basedon the superposition principle of electromagnetism whichstates that the force exerted on a point via other points isinversely proportional to the distance between the points anddirectly proportional to the product of their charges Points insolution space are considered as particles The charge of eachpoint is computed in accordancewith their objective functionvalue In classical physics charge of a particle generallyremains constant but in this heuristic the charge of each pointis not constant and changes from iteration to iteration Thecharge 119902119894 of each point 119894 determines its power of attraction orrepulsion This charge of a particle is evaluated as follows

119902119894= 119890(minus119899((119891(119909

119894)minus119891(119909

best))sum119898

119896=1(119891(119909119896)minus119891(119909

best)))) forall119894 (36)

where 119898 is the total number of points and 119899 is the numberof dimensions This formula shows that points having betterobjective values will possess higher charges This heuristicdoes not use signs to indicate positive or negative chargeas in case of electric charge So direction of force (whetherattractive or repulsive force) is determined by the objectivefunction values (fitness) of two particular points If point119909119895 has better value than 119909119894 then corresponding force is

considered as attractive otherwise repulsive That means that119909best attracts all other points towards it The total force 119865119894

(attractive or repulsive) exerted on point 119894 is computed by thefollowing equation

119865119894=

119898

sum119895 = 119894

(119909119895 minus 119909119894)

119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) lt 119891 (119909119894)

(119909119894 minus 119909119895)119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) ge 119891 (119909119894)

forall119894 (37)

After evaluating the total force vector 119865119894 the point 119894 is movedin the direction of the force with a random step length asgiven in (38) Here RNG is a vector whose componentsdenote the allowed feasible movement towards the upperbound or the lower bound

119909119894= 119909119894+ 120582

119865119894

10038171003817100381710038171198651198941003817100381710038171003817(RNG)

119894 = 1 2 119898

(38)

EM algorithm provides good exploration and exploitationmechanism with computation of charge and force Explo-ration and convergence of EM are controlled by the randomparameter 120582 Exploration is also controlled with RNG bylimiting movements of particles

Debels et al [146] propose a hybrid version of EMcombining the concept of GA with EM

Numerous applications are developed on the basis ofthis algorithm such as scheduling problems [147ndash150] coursetimetabling [151] PID controller [152] fuzzy system [153ndash155] vehicle routing problem [156] networking [157] inven-tory control [158] neural network [159 160] TSP [161 162]feature selection [163] antenna application [164] roboticsapplication [165] flow path designing [166] and vehiclerouting [167]

35 Glass-Demagnetization-Based Algorithms

351 HO HO [18] is inspired by the demagnetization pro-cess of a magnetic sample A magnetic sample comes toa very stable low-energy state called ground state when itis demagnetized by an oscillating magnetic field of slowlydecreasing amplitude After demagnetization the system areshakeup repeatedly to obtain improved result HO simulatesthese two processes of magnetic sample to get low-energystate by repeating demagnetization followed by a number ofshakeups

The process of demagnetization is mainly for explo-ration and convergence After exploring better solutions aresearched by performing a number of shake-up operationsThe algorithm possesses two kinds of stopping conditionsfirstly fixed number of shakeups 119899119904 for each instance of agiven size119873 and secondly required number of shakeups119872reqto obtain the current low-energy state or global optimumBesides this119872req repetition of shakeup in current low-energystateminimumnumber of shakeups 119899119904min is set to ensure that


algorithm does not accept suboptimum too early Similarlythe maximum number of shakeups 119899119904max is also set to avoidwasting time on hard situation HO algorithm is applied toTSP [168] spin glasses [169] vehicle routing problem [170]protein folding [171] and so forth

36 Electrostatics-Based Algorithms

361 CSS This algorithm inherits Coulombrsquos law Gaussrsquoslaw and superposition principle from electrostatics and theNewtonian laws ofmechanics CSS [19] deploys each solutionas a Charged Particle (CP) If two charged particles havingcharges 119902119894 and 119902119895 reside at distance 119903119894119895 then according toCoulombrsquos law electric force 119865119894119895 exerted between them is asfollows

119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)

Here 119896119890 is a constant called the Coulomb constant Now if 119902119894amount of charges is uniformly distributed within a sphere ofradius ldquo119886rdquo then electric field 119864119894119895 at a point outside the sphereis as follows

119864119894119895 = 119896119890119902119894

1199032119894119895

(40)

The electric field at a point inside the sphere can be obtainedusing Gaussrsquos law as follows

119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)

The resultant force on a charge 119902119895 at position 119903119895 due to theelectric field of a charge 119902119894 at position 119903119894 can be expressed invector form as

119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)

Formultiple charged particles this equation can be expressedas follows

119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)

times119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)

To formulate the concept of charges into CSS algorithm a setof charged particles are considered Each point in the solutionspace is considered as possible positions of any chargedparticle Charges in each charged particle is computed asfollows

119902119894 =fit (119894) minus fitworstfirbest minus fitworst

(44)

Distance among particles is computed with the following

119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)

Radius of particle is computed with

119886 = 010 timesmax (119909119894max minus 119909119894min | 119894 = 1 2 119899) (46)

The value of the resultant electrical force acting on a chargedparticle is determined as follows

119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)

Here 119875119894119895 defines the attractiveness or repulsiveness of theforce exerted A good particle may attract a bad one andsimilarly bad one can also attract good one So if bad oneattracts good one then it is not suitable for an optimizationproblem The parameter 119875119894119895 limits these kinds of attractionsas follows

119875119894119895 =

1if (fit (119894) minus fitbest

fit (119895) minus fit (119894)gt rand)

or (fit (119895) gt fit (119894))0 otherwise

(48)

Again in Newtonian mechanics or classical mechanics thevelocity V of a particle is defined as follows

V =119903new minus 119903old119905new minus 119905old

=119903new minus 119903old

Δ119905 (49)

Displacement from 119903old to 119903new position along with accelera-tion 119886 can be expressed as follows

119903new =1

2119886 sdot Δ1199052+ Vold sdot Δ119905 + 119903old (50)

Newtonrsquos second law states that ldquothe acceleration of an objectis directly proportional to the net force acting on it andinversely proportional to its massrdquo that is 119865 = 119898119886 so 119903newcan be expressed as follows

119903new =1

2

119865

119898sdot Δ1199052+ Vold sdot Δ119905 + 119903old (51)

In CSS movements due to the electric force exerted amongthose particles are measured and accordingly new positionsof particles are updated New position (119883119895new) of CP andwith which velocity (119881119895new) will reach the position (119883119895new)is computed as follows

119883119895new = rand1198951 sdot 119896119886 sdot119865119895

119898119895sdot Δ1199052+ rand1198952 sdot 119896V sdot Δ119905 + 119883119895old

119881119895new =119883119895new minus 119883119895old

Δ119905

(52)

Here 119896119886 is the parameter related to the attracting forces and119896V is velocity coefficient The effect of the pervious velocityand the resultant force acting on a charged particle can be


decreased with parameter 119896V or increased with parameter 119896119886These parameters can be computed as follows

119896V = 05 (1 minusiter

itermax)

119896119886 = 05 (1 +iter

itermax)

(53)

where iter is the current iteration number and itermax is themaximum number of iterations

The CSS algorithm possesses good exploring as well asexploiting capability of solution domain Exploitation of CPis mainly ensured by the resulting electric force 119865119895 of anyparticle 119895 Handling of attractiveness and repulsiveness ofresulting force of any CPwith the noble concept of parameter119875119894119895 is very effective for exploitation However whether CPis going to explore or exploit the search space depends onthe parameters 119896119886 and 119896V Higher value of 119896119886 implies higherimpact on resulting electric force which results exploitationof search space Whereas higher value of 119896V implies highexploration Initially values of 119896119886 and 119896V are almost same butgradually 119896119886 increases and 119896V decreases Hence at the begin-ning the algorithmexplores the search space As in successiveiterations 119896119886 increases gradually the effect of attraction ofgood solutions also increases Thus the algorithm ensuresconvergence towards better solutionsThe algorithmdoes notsuffer from premature convergence due high exploration atthe beginning of the algorithm However since good solutionattracts others if initial set of CPs not uniformly distributedover solution space then the algorithm may be trapped intoany local optima

Applications of this algorithm are mainly related tostructural engineering designs [172ndash175] and geometry opti-mization [176]

4 Conclusion

In this paper we have categorically discussed various opti-mization algorithms that are mainly inspired by physicsMajor areas covered by these algorithms are quantum theoryelectrostatics electromagnetism Newtonrsquos gravitational lawand laws of motion This study shows that most of thesealgorithms are inspired by quantum computing and signifi-cant numbers of applications are developed on the basis ofthem Parallel nature of quantum computing perhaps attractsresearchers towards quantum-based algorithms Anothermost attractive area of physics for inspiration is Newtonrsquosgravitational laws and laws of motion We have realized thathybridization of quantum computing and biological phe-nomenon drawsmost attention these days As biological phe-nomenon suggests best strategies and quantum computingprovide simultaneity to those strategies so merging of bothinto one implies better result In this paper we have studiedformational aspects of all the major algorithms inspired byphysics We hope this study will definitely be beneficialfor new researchers and motivate them to formulate greatsolutions from those inspirational theorems of physics tooptimization problems

Abbreviations

ACO Ant colony optimizationAPO Artificial physics optimizationBB-BC Big bang-big crunchBFO Bacterial forging optimizationBGSA Binary gravitational search algorithmBIS Biological immune systemBQEA Binary Quantum-inspired evolutionary

algorithmCFO Central force optimizationCQACO Continuous quantum ant colony

optimizationCSS Charged system searchEAPO Extended artificial physics optimizationECFO Extended central force optimizationEM Electromagnetism-like heuristicGA Genetic AlgorithmGbSA Galaxy-based search algorithmGIO Gravitational interaction optimizationGSA Gravitational search algorithmHO Hysteretic optimizationHQGA Hybrid quantum-inspired genetic algorithmHS Harmony searchIGOA Immune gravitation inspired optimization

algorithmIQEA Improved quantum evolutionary algorithmLP Linear programmingMOGSA Multiobjective gravitational search algorithmNLP Nonlinear programmingPSO Particle swarm optimizationPSOGSA PSO gravitational search algorithmQBSO Quantum-inspired bacterial swarming

optimizationQEA Quantum-inspired evolutionary algorithmQGA Quantum-inspired genetic algorithmQGO Quantum genetic optimizationQICA Quantum-inspired immune clonal algorithmQPSO Quantum-behaved particle swarm

optimizationQSE Quantum swarm evolutionary algorithmRQGA Reduced quantum genetic algorithmSA Simulated annealingTSP Travelling salesman problemUBB-CBC Unified big bang-chaotic big crunchVM-APO Vector model of artificial physics

optimizationvQEA Versatile quantum-inspired evolutionary

algorithm

References

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[2] D P Bertsekas Nonlinear Programmingby Athena ScientificBelmont Mass USA 2nd edition 1999

[3] J H Holland ldquoGenetic algorithms and the optimal allocationof trialsrdquo SIAM Journal on Computing vol 2 no 2 pp 88ndash1051973


[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks IV pp 1942ndash1948 December 1995

[5] S Kirkpatrick and M P Vecchi ldquoOptimization by simulatedannealingrdquo Science vol 220 no 4598 pp 671ndash680 1983

[6] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[7] R P Feynman ldquoSimulating physics with computersrdquo Interna-tional Journal ofTheoretical Physics vol 21 no 6-7 pp 467ndash4881982

[8] R P Feynman ldquoQuantummechanical computersrdquo Foundationsof Physics vol 16 no 6 pp 507ndash531 1986

[9] A Narayanan and M Moore ldquoQuantum-inspired genetic algo-rithmsrdquo in Proceedings of the IEEE International Conference onEvolutionary Computation (ICEC rsquo96) pp 61ndash66 May 1996

[10] J SunWXu andB Feng ldquoA global search strategy of quantum-behaved particle swarm optimizationrdquo in Proceedings of the2004 IEEE Conference on Cybernetics and Intelligent Systemsvol 1 pp 111ndash116 December 2004

[11] Y Wang X Feng Y Huang et al ldquoA novel quantum swarmevolutionary algorithm and its applicationsrdquo Neurocomputingvol 70 no 4ndash6 pp 633ndash640 2007

[12] S I Birbil and S Fang ldquoAn electromagnetism-like mechanismfor global optimizationrdquo Journal of Global Optimization vol 25no 3 pp 263ndash282 2003

[13] O K Erol and I Eksin ldquoA new optimizationmethod Big Bang-Big CrunchrdquoAdvances in Engineering Software vol 37 no 2 pp106ndash111 2006

[14] R A Formato ldquoCentral force optimization a newmetaheuristicwith applications in applied electromagneticsrdquo Progress inElectromagnetics Research vol 77 pp 425ndash491 2007

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] L Xie J Zeng and Z Cui ldquoGeneral framework of artificialphysics optimization algorithmrdquo in Proceedings of the WorldCongress on Nature and Biologically Inspired Computing (NaBICrsquo09) pp 1321ndash1326 IEEE December 2009

[17] J Flores R Lopez and J Barrera ldquoGravitational interactionsoptimizationrdquo in Learning and Intelligent Optimization pp226ndash237 Springer Berlin Germany 2011

[18] K F Pal ldquoHysteretic optimization for the Sherrington-Kirkpatrick spin glassrdquo Physica A vol 367 pp 261ndash268 2006

[19] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3 pp 267ndash289 2010

[20] H Shah-Hosseini ldquoPrincipal components analysis by thegalaxy-based search algorithm a novel metaheuristic for con-tinuous optimisationrdquo International Journal of ComputationalScience and Engineering vol 6 no 1-2 pp 132ndash140 2011

[21] L Jiao Y Li M Gong and X Zhang ldquoQuantum-inspiredimmune clonal algorithm for global optimizationrdquo IEEE Trans-actions on Systems Man and Cybernetics B vol 38 no 5 pp1234ndash1253 2008

[22] W Li Q Yin and X Zhang ldquoContinuous quantum ant colonyoptimization and its application to optimization and analysisof induction motor structurerdquo in Proceedings of the IEEE 5thInternational Conference on Bio-Inspired Computing Theoriesand Applications (BIC-TA rsquo10) pp 313ndash317 September 2010

[23] Y Zhang L Wu Y Zhang and J Wang ldquoImmune gravitationinspired optimization algorithmrdquo in Advanced Intelligent Com-puting pp 178ndash185 Springer Berlin Germany 2012

[24] C Jinlong andHGao ldquoA quantum-inspired bacterial swarmingoptimization algorithm for discrete optimization problemsrdquo inAdvances in Swarm Intelligence pp 29ndash36 Springer BerlinGermany 2012

[25] D Ding D Qi X Luo J Chen X Wang and P Du ldquoConver-gence analysis and performance of an extended central forceoptimization algorithmrdquo Applied Mathematics and Computa-tion vol 219 no 4 pp 2246ndash2259 2012

[26] R C Green II L Wang and M Alam ldquoTraining neuralnetworks using central force optimization and particle swarmoptimization insights and comparisonsrdquo Expert Systems withApplications vol 39 no 1 pp 555ndash563 2012

[27] R A Formato ldquoCentral force optimization applied to the PBMsuite of antenna benchmarksrdquo 2010 httparxivorgabs10030221

[28] G M Qubati R A Formato and N I Dib ldquoAntenna bench-mark performance and array synthesis using central forceoptimisationrdquo IET Microwaves Antennas and Propagation vol4 no 5 pp 583ndash592 2010

[29] D F Spears W Kerr W Kerr and S Hettiarachchi ldquoAnoverview of physicomimeticsrdquo in Swarm Robotics vol 3324 ofLecture Notes in Computer Science State of the Art pp 84ndash97Springer Berlin Germany 2005

[30] L Xie and J Zeng ldquoAn extended artificial physics optimizationalgorithm for global optimization problemsrdquo in Proceedingsof the 4th International Conference on Innovative ComputingInformation and Control (ICICIC rsquo09) pp 881ndash884 December2009

[31] L Xie J Zeng andZCui ldquoThe vectormodel of artificial physicsoptimization algorithm for global optimization problemsrdquo inIntelligent Data Engineering and Automated LearningmdashIDEAL2009 pp 610ndash617 Springer Berlin Germany 2009

[32] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoBGSAbinary gravitational search algorithmrdquo Natural Computing vol9 no 3 pp 727ndash745 2010

[33] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 July 2010

[34] S Mirjalili and S Z M Hashim ldquoA new hybrid PSOGSAalgorithm for function optimizationrdquo in Proceedings of the Inter-national Conference on Computer and Information Application(ICCIA rsquo10) pp 374ndash377 December 2010

[35] E Rashedi H Nezamabadi-Pour S Saryazdi and M FarsangildquoAllocation of static var compensator using gravitational searchalgorithmrdquo in Proceedings of the 1st Joint Congress on Fuzzy andIntelligent Systems pp 29ndash31 2007

[36] B Shaw V Mukherjee and S P Ghoshal ldquoA novel opposition-based gravitational search algorithm for combined economicand emission dispatch problems of power systemsrdquo Interna-tional Journal of Electrical Power and Energy Systems vol 35no 1 pp 21ndash33 2012

[37] S Duman U Guvenc Y Sonmez and N Yorukeren ldquoOptimalpower flow using gravitational search algorithmrdquo Energy Con-version and Management vol 59 pp 86ndash95 2012

[38] P Purwoharjono M Abdillah O Penangsang and A Soepri-janto ldquoVoltage control on 500 kV Java-Bali electrical powersystem for power lossesminimization using gravitational search


algorithmrdquo in Proceedings of the 1st International Conference onInformatics and Computational Intelligence (ICI rsquo11) pp 11ndash17December 2011

[39] S Duman Y Soonmez U Guvenc andN Yorukeren ldquoOptimalreactive power dispatch using a gravitational search algorithmrdquoIET Generation Transmission amp Distribution vol 6 no 6 pp563ndash576 2012

[40] S Mondal A Bhattacharya and S Halder ldquoSolution of costconstrained emission dispatch problems considering windpower generation using gravitational search algorithmrdquo inProceedings of the International Conference on Advances inEngineering Science and Management (ICAESM rsquo12) pp 169ndash174 IEEE 2012

[41] A Bhattacharya and P K Roy ldquoSolution of multi-objectiveoptimal power flow using gravitational search algorithmrdquo IETGeneration Transmission amp Distribution vol 6 no 8 pp 751ndash763 2012

[42] S Duman Y Sonmez U Guvenc and N Yorukeren ldquoAppli-cation of gravitational search algorithm for optimal reactivepower dispatch problemrdquo in Proceedings of the InternationalSymposium on Innovations in Intelligent Systems and Applica-tions (INISTA rsquo11) pp 1ndash5 IEEE June 2011

[43] S Duman U Guvenc and N Yurukeren ldquoGravitational searchalgorithm for economic dispatch with valve-point effectsrdquoInternational Review of Electrical Engineering vol 5 no 6 pp2890ndash2895 2010

[44] S Duman A B Arsoy and N Yorukeren ldquoSolution of eco-nomic dispatch problem using gravitational search algorithmrdquoin Proceedings of the 7th International Conference on Electricaland Electronics Engineering (ELECO rsquo11) pp I54ndashI59 December2011

[45] MGhalambaz A R NoghrehabadiM A Behrang E AssarehA Ghanbarzadeh and N Hedayat ldquoA Hybrid Neural Networkand Gravitational Search Algorithm (HNNGSA) method tosolve well known Wessingerrsquos equationrdquo World Academy ofScience Engineering and Technology vol 73 pp 803ndash807 2011

[46] R Precup R David E M Petriu S Preitl and M RadacldquoGravitational search algorithm-based tuning of fuzzy controlsystems with a reduced parametric sensitivityrdquo in Soft Com-puting in Industrial Applications pp 141ndash150 Springer BerlinGermany 2011

[47] R Precup R David E M Petriu S Preitl and M RadacldquoFuzzy control systems with reduced parametric sensitivitybased on simulated annealingrdquo IEEE Transactions on IndustrialElectronics vol 59 no 8 pp 3049ndash3061 2012

[48] M A Behrang E Assareh M Ghalambaz M R Assari and AR Noghrehabadi ldquoForecasting future oil demand in Iran usingGSA (Gravitational Search Algorithm)rdquo Energy vol 36 no 9pp 5649ndash5654 2011

[49] M Khajehzadeh M R Taha A El-Shafie and M EslamildquoA modified gravitational search algorithm for slope stabilityanalysisrdquo Engineering Applications of Artificial Intelligence vol25 8 pp 1589ndash1597 2012

[50] A Hatamlou S Abdullah and H Nezamabadi-Pour ldquoAppli-cation of gravitational search algorithm on data clusteringrdquo inRough Sets and Knowledge Technology pp 337ndash346 SpringerBerlin Germany 2011

[51] M Yin Y Hu F Yang X Li and W Gu ldquoA novel hybrid K-harmonic means and gravitational search algorithm approachfor clusteringrdquo Expert Systems with Applications vol 38 no 8pp 9319ndash9324 2011

[52] C Li J Zhou B Fu P Kou and J Xiao ldquoT-S fuzzy modelidentification with a gravitational search-based hyperplaneclustering algorithmrdquo IEEE Transactions on Fuzzy Systems vol20 no 2 pp 305ndash317 2012

[53] A Bahrololoum H Nezamabadi-Pour H Bahrololoum andM Saeed ldquoA prototype classifier based on gravitational searchalgorithmrdquo Applied Soft Computing Journal vol 12 no 2 pp819ndash825 2012

[54] J P Papa A Pagnin S A Schellini et al ldquoFeature selectionthrough gravitational search algorithmrdquo in Proceedings of the36th IEEE International Conference on Acoustics Speech andSignal Processing (ICASSP rsquo11) pp 2052ndash2055 May 2011

[55] B Zibanezhad K Zamanifar N Nematbakhsh and F Mar-dukhi ldquoAn approach for web services composition based onQoS and gravitational search algorithmrdquo in Proceedings ofthe International Conference on Innovations in InformationTechnology (IIT rsquo09) pp 340ndash344 IEEE December 2009

[56] S Duman D Maden and U Guvenc ldquoDetermination of thePID controller parameters for speed and position control ofDC motor using gravitational search algorithmrdquo in Proceedingsof the 7th International Conference on Electrical and ElectronicsEngineering (ELECO rsquo11) pp I225ndashI229 IEEE December 2011

[57] W X Gu X T Li L Zhu et al ldquoA gravitational search algorithmfor flow shop schedulingrdquo CAAI Transaction on IntelligentSystems vol 5 no 5 pp 411ndash418 2010

[58] D Hoffman ldquoA brief overview of the biological immunesystemrdquo 2011 httpwwwhealthynet

[59] M Cleric and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[60] M S Innocente and J Sienz ldquoParticle swarm optimization withinertia weight and constriction factorrdquo in Proceedings of theInternational conference on swarm intelligence (ICSI rsquo11) 2011

[61] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[62] M Udrescu L Prodan and M Vladutiu ldquoImplementingquantum genetic algorithms a solution based on Groverrsquosalgorithmrdquo in Proceedings of the 3rd Conference on ComputingFrontiers (CF rsquo06) pp 71ndash81 ACM May 2006

[63] B Li and L Wang ldquoA hybrid quantum-inspired genetic algo-rithm for multiobjective flow shop schedulingrdquo IEEE Transac-tions on Systems Man and Cybernetics B vol 37 no 3 pp 576ndash591 2007

[64] L Wang F Tang and H Wu ldquoHybrid genetic algorithmbased on quantum computing for numerical optimization andparameter estimationrdquo Applied Mathematics and Computationvol 171 no 2 pp 1141ndash1156 2005

[65] A Malossini and T Calarco ldquoQuantum genetic optimizationrdquoIEEE Transactions on Evolutionary Computation vol 12 no 2pp 231ndash241 2008

[66] A Layeb S Meshoul and M Batouche ldquoquantum geneticalgorithm for multiple RNA structural alignmentrdquo in Proceed-ings of the 2nd Asia International Conference on Modelling andSimulation (AIMS rsquo08) pp 873ndash878 May 2008

[67] D Chang and Y Zhao ldquoA dynamic niching quantum geneticalgorithm for automatic evolution of clustersrdquo in Proceedingsof the 14th International Conference on Computer Analysis ofImages and Patterns vol 2 pp 308ndash315 2011


[68] J Xiao Y Yan Y Lin L Yuan and J Zhang ldquoA quantum-inspired genetic algorithm for data clusteringrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (CEC rsquo08)pp 1513ndash1519 June 2008

[69] H Talbi A Draa and M Batouche ldquoA new quantum-inspiredgenetic algorithm for solving the travelling salesman problemrdquoin Proceedings of the IEEE International Conference on IndustrialTechnology (ICIT rsquo04) vol 3 pp 1192ndash1197 December 2004

[70] K-H Han K-H Park C-H Lee and J-H Kim ldquoParallelquantum-inspired genetic algorithm for combinatorial opti-mization problemrdquo in Proceedings of the 2001 Congress onEvolutionary Computation vol 2 pp 1422ndash1429 IEEE May2001

[71] L Yan H Chen W Ji Y Lu and J Li ldquoOptimal VSM modeland multi-object quantum-inspired genetic algorithm for webinformation retrievalrdquo in Proceedings of the 1st InternationalSymposium on Computer Network and Multimedia Technology(CNMT rsquo09) pp 1ndash4 IEEE December 2009

[72] Z Mo G Wu Y He and H Liu ldquoquantum genetic algorithmfor scheduling jobs on computational gridsrdquo in Proceedingsof the International Conference on Measuring Technology andMechatronics Automation (ICMTMA rsquo10) pp 964ndash967 March2010

[73] Y Zhang J Liu Y Cui X Hei and M Zhang ldquoAn improvedquantum genetic algorithm for test suite reductionrdquo in Proceed-ings of the IEEE International Conference on Computer Scienceand Automation Engineering (CSAE rsquo11) pp 149ndash153 June 2011

[74] J Lee W Lin G Liao and T Tsao ldquoquantum genetic algorithmfor dynamic economic dispatch with valve-point effects andincluding wind power systemrdquo International Journal of Electri-cal Power and Energy Systems vol 33 no 2 pp 189ndash197 2011

[75] J Dai and H Zhang ldquoA novel quantum genetic algorithm forarea optimization of FPRM circuitsrdquo in Proceedings of the 3rdInternational Symposium on Intelligent Information TechnologyApplication (IITA 09) pp 408ndash411 November 2009

[76] L Chuang Y Chiang and C Yang ldquoA quantum genetic algo-rithm for operon predictionrdquo in Proceedings of the IEEE 26thInternational Conference on Advanced Information Networkingand Applications (AINA rsquo12) pp 269ndash275 March 2012

[77] H Xing X Liu X Jin L Bai and Y Ji ldquoA multi-granularityevolution based quantum genetic algorithm for QoS multicastrouting problem in WDM networksrdquo Computer Communica-tions vol 32 no 2 pp 386ndash393 2009

[78] W Luo ldquoA quantum genetic algorithm based QoS routingprotocol for wireless sensor networksrdquo in Proceedings of theIEEE International Conference on Software Engineering andService Sciences (ICSESS rsquo10) pp 37ndash40 IEEE July 2010

[79] J Wang and R Zhou ldquoA novel quantum genetic algorithmfor PID controllerrdquo in Proceedings of the 6th InternationalConference on Advanced Intelligent Computing Theories andApplications Intelligent Computing pp 72ndash77 2010

[80] BHan J Jiang YGao and JMa ldquoA quantumgenetic algorithmto solve the problem of multivariaterdquo Communications inComputer and Information Science vol 243 no 1 pp 308ndash3142011

[81] Y Zheng J Liu W Geng and J Yang ldquoQuantum-inspiredgenetic evolutionary algorithm for course timetablingrdquo inProceedings of the 3rd International Conference on Genetic andEvolutionary Computing (WGEC rsquo09) pp 750ndash753 October2009

[82] Y J Lv and N X Liu ldquoApplication of quantum geneticalgorithm on finding minimal reductrdquo in Proceedings of the

IEEE International Conference on Granular Computing (GRCrsquo07) pp 728ndash733 November 2007

[83] X J Zhang S Li Y Shen and S M Song ldquoEvaluation ofseveral quantum genetic algorithms in medical image regis-tration applicationsrdquo in Proceedings of the IEEE InternationalConference on Computer Science and Automation Engineering(CSAE rsquo12) vol 2 pp 710ndash713 IEEE 2012

[84] H Talbi A Draa and M Batouche ldquoA new quantum-inspiredgenetic algorithm for solving the travelling salesman problemrdquoin Proceedings of the IEEE International Conference on IndustrialTechnology (ICIT rsquo04) pp 1192ndash1197 December 2004

[85] S Bhattacharyya and S Dey ldquoAn efficient quantum inspiredgenetic algorithm with chaotic map model based interferenceand fuzzy objective function for gray level image thresholdingrdquoin Proceedings of the International Conference on ComputationalIntelligence and Communication Systems (CICN rsquo11) pp 121ndash125IEEE October 2011

[86] K Benatchba M Koudil Y Boukir and N Benkhelat ldquoImagesegmentation using quantum genetic algorithmsrdquo in Proceed-ings of the 32nd Annual Conference on IEEE Industrial Electron-ics (IECON rsquo06) pp 3556ndash3562 IEEE November 2006

[87] M Liu C Yuan and T Huang ldquoA novel real-coded quantumgenetic algorithm in radiation pattern synthesis for smartantennardquo in Proceedings of the IEEE International Conferenceon Robotics and Biomimetics (ROBIO rsquo07) pp 2023ndash2026 IEEEDecember 2007

[88] R Popa V Nicolau and S Epure ldquoA new quantum inspiredgenetic algorithm for evolvable hardwarerdquo in Proceedings ofthe 3rd International Symposium on Electrical and ElectronicsEngineering (ISEEE rsquo10) pp 64ndash69 September 2010

[89] H Yu and J Fan ldquoParameter optimization based on quantumgenetic algorithm for generalized fuzzy entropy thresholdingsegmentation methodrdquo in Proceedings of the 5th InternationalConference on Fuzzy Systems and Knowledge Discovery (FSKDrsquo08) vol 1 pp 530ndash534 IEEE October 2008

[90] P C Shill M F Amin M A H Akhand and KMurase ldquoOpti-mization of interval type-2 fuzzy logic controller using quantumgenetic algorithmsrdquo in Proceedings of the IEEE InternationalConference on Fuzzy Systems (FUZZ-IEEE rsquo12) pp 1ndash8 June2012

[91] M Cao and F Shang ldquoTraining of process neural networksbased on improved quantum genetic algorithmrdquo in Proceedingsof theWRIWorld Congress on Software Engineering (WCSE rsquo09)vol 2 pp 160ndash165 May 2009

[92] Y Sun and M Ding ldquoquantum genetic algorithm for mobilerobot path planningrdquo in Proceedings of the 4th InternationalConference on Genetic and Evolutionary Computing (ICGECrsquo10) pp 206ndash209 December 2010

[93] K Han and J Kim ldquoQuantum-inspired evolutionary algorithmfor a class of combinatorial optimizationrdquo IEEE Transactions onEvolutionary Computation vol 6 no 6 pp 580ndash593 2002

[94] R Zhang and H Gao ldquoImproved quantum evolutionary algo-rithm for combinatorial optimization problemrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics (ICMLC rsquo07) vol 6 pp 3501ndash3505 August 2007

[95] M D Platel S Sehliebs and N Kasabov ldquoA versatile quantum-inspired evolutionary algorithmrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo07) pp 423ndash430September 2007

[96] P Li and S Li ldquoQuantum-inspired evolutionary algorithm forcontinuous space optimization based on Bloch coordinates ofqubitsrdquo Neurocomputing vol 72 no 1ndash3 pp 581ndash591 2008



[98] P Mahdabi S Jalili and M Abadi ldquoA multi-start quantum-inspired evolutionary algorithm for solving combinatorial opti-mization problemsrdquo in Proceedings of the 10th Annual Geneticand Evolutionary Computation Conference (GECCO rsquo08) pp613ndash614 ACM July 2008

[99] H Talbi M Batouche and A Draao ldquoA quantum-inspiredevolutionary algorithm formultiobjective image segmentationrdquoInternational Journal of Mathematical Physical and EngineeringSciences vol 1 no 2 pp 109ndash114 2007

[100] Y Kim J Kim and K Han ldquoQuantum-inspired multiobjec-tive evolutionary algorithm for multiobjective 01 knapsackproblemsrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo06) pp 2601ndash2606 July 2006

[101] A Narayan and C Patvardhan ldquoA novel quantum evolutionaryalgorithm for quadratic knapsack problemrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo09) pp 1388ndash1392 October 2009

[102] A R Hota and A Pat ldquoAn adaptive quantum-inspired dif-ferential evolution algorithm for 0-1 knapsack problemrdquo inProceedings of the 2ndWorld Congress onNature andBiologicallyInspired Computing (NaBIC rsquo10) pp 703ndash708 December 2010

[103] Y Ji and H Xing ldquoA memory storable quantum inspired evolu-tionary algorithm for network coding resource minimizationrdquoin Evolutionary Algorithms InTech Shanghai China 2011

[104] H Xing Y Ji L Bai and Y Sun ldquoAn improved quantum-inspired evolutionary algorithm for coding resource optimiza-tion based network coding multicast schemerdquo InternationalJournal of Electronics and Communications vol 64 no 12 pp1105ndash1113 2010

[105] A da Cruz M M B R Vellasco and M Pacheco ldquoQuantum-inspired evolutionary algorithm for numerical optimizationrdquoin Hybrid Evolutionary Algorithms pp 19ndash37 Springer BerlinGermany 2007

[106] G Zhang and H Rong ldquoReal-observation quantum-inspiredevolutionary algorithm for a class of numerical optimizationproblemsrdquo in Proceedings of the 7th international conference onComputational Science Part IV (ICCS rsquo07) vol 4490 pp 989ndash996 2007

[107] R Setia and K H Raj ldquoQuantum inspired evolutionary algo-rithm for optimization of hot extrusion processrdquo InternationalJournal of Soft Computing and Engineering vol 2 no 5 p 292012

[108] T Lau Application of quantum-inspired evolutionary algorithmin solving the unit commitment problem [dissertation]TheHongKong Polytechnic University Hong Kong 2011

[109] C Y Chung H Yu and K P Wong ldquoAn advanced quantum-inspired evolutionary algorithm for unit commitmentrdquo IEEETransactions on Power Systems vol 26 no 2 pp 847ndash854 2011

[110] J G Vlachogiannis and K Y Lee ldquoQuantum-inspired evolu-tionary algorithm for real and reactive power dispatchrdquo IEEETransactions on Power Systems vol 23 no 4 pp 1627ndash16362008

[111] U Pareek M Naeem and D C Lee ldquoQuantum inspired evolu-tionary algorithm for joint user selection and power allocationfor uplink cognitive MIMO systemsrdquo in Proceedings of the IEEESymposium on Computational Intelligence in Scheduling (SCISrsquo11) pp 33ndash38 April 2011

[112] J Chen ldquoApplication of quantum-inspired evolutionary algo-rithm to reduce PAPR of anOFDMsignal using partial transmit

sequences techniquerdquo IEEE Transactions on Broadcasting vol56 no 1 pp 110ndash113 2010

[113] J Jang K Han and J Kim ldquoFace detection using quantum-inspired evolutionary algorithmrdquo in Proceedings of the 2004Congress on Evolutionary Computation (CEC rsquo04) vol 2 pp2100ndash2106 June 2004

[114] J Jang K Han and J Kim ldquoQuantum-inspired evolutionaryalgorithm-based face verificationrdquo in Genetic and EvolutionaryComputationmdashGECCO 2003 pp 214ndash214 Springer BerlinGermany 2003

[115] K Fan A Brabazon C OrsquoSullivan andM OrsquoNeill ldquoQuantum-inspired evolutionary algorithms for financial data analysisrdquo inApplications of Evolutionary Computing pp 133ndash143 SpringerBerlin Germany 2008

[116] K Fan A Brabazon C OrsquoSullivan and M OrsquoNeill ldquoOptionpricingmodel calibration using a real-valued quantum-inspiredevolutionary algorithmrdquo in Proceedings of the 9th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo07) pp 1983ndash1990 ACM July 2007

[117] K Fan A Brabazon C OSullivan and M ONeill ldquoQuantum-inspired evolutionary algorithms for calibration of the VGoption pricing modelrdquo in Applications of Evolutionary Comput-ing pp 189ndash198 Springer Berlin Germany 2007

[118] R A de Araujo ldquoA quantum-inspired evolutionary hybridintelligent approach for stock market predictionrdquo InternationalJournal of Intelligent Computing and Cybernetics vol 3 no 1 pp24ndash54 2010

[119] Z Huang Y Wang C Yang and C Wu ldquoA new improvedquantum-behaved particle swarm optimization modelrdquo in Pro-ceedings of the 4th IEEE Conference on Industrial Electronics andApplications (ICIEA rsquo09) pp 1560ndash1564 May 2009

[120] J Chang F An and P Su ldquoA quantum-PSO algorithm forno-wait flow shop scheduling problemrdquo in Proceedings of theChinese Control and Decision Conference (CCDC rsquo10) pp 179ndash184 May 2010

[121] X Wu B Zhang K Wang J Li and Y Duan ldquoA quantum-inspired Binary PSO algorithm for unit commitment with windfarms considering emission reductionrdquo in Proceedings of theInnovative Smart Grid TechnologiesmdashAsia (ISGT rsquo12) pp 1ndash6IEEE May 2012

[122] Y Jeong J Park S Jang and K Y Lee ldquoA new quantum-inspired binary PSO for thermal unit commitment problemsrdquoin Proceedings of the 15th International Conference on IntelligentSystem Applications to Power Systems (ISAP rsquo09) pp 1ndash6November 2009

[123] H N A Hamed N Kasabov and S M Shamsuddin ldquoInte-grated feature selection and parameter optimization for evolv-ing spiking neural networks using quantum inspired particleswarm optimizationrdquo in Proceedings of the International Confer-ence on Soft Computing and Pattern Recognition (SoCPaR rsquo09)pp 695ndash698 December 2009

[124] A A Ibrahim A Mohamed H Shareef and S P GhoshalldquoAn effective power qualitymonitor placementmethod utilizingquantum-inspired particle swarm optimizationrdquo in Proceedingsof the International Conference on Electrical Engineering andInformatics (ICEEI rsquo11) pp 1ndash6 July 2011

[125] F Yao Z Y Dong K Meng Z Xu H H Iu and K WongldquoQuantum-inspired particle swarm optimization for power sys-temoperations consideringwind power uncertainty and carbontax in Australiardquo IEEE Transactions on Industrial Informaticsvol 8 no 4 pp 880ndash888 2012


[126] Z Zhisheng ldquoQuantum-behaved particle swarm optimizationalgorithm for economic load dispatch of power systemrdquo ExpertSystems with Applications vol 37 no 2 pp 1800ndash1803 2010

[127] A Chen G Yang and Z Wu ldquoHybrid discrete particleswarm optimization algorithm for capacitated vehicle routingproblemrdquo Journal of Zhejiang University vol 7 no 4 pp 607ndash614 2006

[128] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers and Operations Research vol 36 no5 pp 1693ndash1702 2009

[129] Y Marinakis M Marinaki and G Dounias ldquoA hybrid particleswarm optimization algorithm for the vehicle routing problemrdquoEngineering Applications of Artificial Intelligence vol 23 no 4pp 463ndash472 2010

[130] S N Omkar R Khandelwal T V S Ananth G Narayana Naikand S Gopalakrishnan ldquoQuantum behaved Particle SwarmOptimization (QPSO) for multi-objective design optimizationof composite structuresrdquo Expert Systems with Applications vol36 no 8 pp 11312ndash11322 2009

[131] L D S Coelho ldquoGaussian quantum-behaved particle swarmoptimization approaches for constrained engineering designproblemsrdquo Expert Systems with Applications vol 37 no 2 pp1676ndash1683 2010

[132] M Ykhlef ldquoA quantum swarm evolutionary algorithm formining association rules in large databasesrdquo Journal of KingSaud University vol 23 no 1 pp 1ndash6 2011

[133] MDorigo and T StiitzleAnt Colony Optimization pp 153ndash222chapter 4 MIT Press Cambridge Mass USA 1st edition 2004

[134] LWang Q Niu andM Fei ldquoA novel quantum ant colony opti-mization algorithmrdquo in Bio-Inspired Computational Intelligenceand Applications pp 277ndash286 Springer Berlin Germany 2007

[135] X You S Liu and Y Wang ldquoQuantum dynamic mechanism-based parallel ant colony optimization algorithmrdquo InternationalJournal of Computational Intelligence Systems vol 3 no 1 pp101ndash113 2010

[136] L Wang Q Niu and M Fei ldquoA novel quantum ant colonyoptimization algorithm and its application to fault diagnosisrdquoTransactions of the Institute of Measurement and Control vol30 no 3-4 pp 313ndash329 2008

[137] Z Yu L Shuhua F Shuai and W Di ldquoA quantum-inspired antcolony optimization for robot coalition formationrdquo in ChineseControl and Decision Conference (CCDC rsquo09) pp 626ndash631 June2009

[138] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[139] T Kumbasar I Eksin M Guzelkaya and E Yesil ldquoBig bangbig crunch optimization method based fuzzy model inversionrdquoinMICAI 2008 Advances in Artificial Intelligence pp 732ndash740Springer Berlin Germany 2008

[140] T Kumbasar E Yesil I Eksin and M Guzelkaya ldquoInversefuzzy model control with online adaptation via big bang-bigcrunch optimizationrdquo in 2008 3rd International Symposium onCommunications Control and Signal Processing (ISCCSP rsquo08)pp 697ndash702 March 2008

[141] M Aliasghary I Eksin and M Guzelkaya ldquoFuzzy-slidingmodel reference learning control of inverted pendulumwithBigBang-Big Crunch optimization methodrdquo in Proceedings of the11th International Conference on Intelligent Systems Design andApplications (ISDA rsquo11) pp 380ndash384 November 2011

[142] H M Genc I Eksin and O K Erol ldquoBig Bang-Big Crunchoptimization algorithm hybridized with local directionalmovesand application to target motion analysis problemrdquo in Proceed-ings of the IEEE International Conference on Systems Man andCybernetics (SMC rsquo10) pp 881ndash887 October 2010

[143] H M Genc and A K Hocaoglu ldquoBearing-only target trackingbased on Big Bang-Big Crunch algorithmrdquo in Proceedings of the3rd International Multi-Conference on Computing in the GlobalInformation Technology (ICCGI rsquo08) pp 229ndash233 July 2008

[144] P Prudhvi ldquoA complete copper optimization technique usingBB-BC in a smart home for a smarter grid and a comparisonwith GArdquo in Proceedings of the 24th Canadian Conference onElectrical and Computer Engineering (CCECE rsquo11) pp 69ndash72May 2011

[145] GM Jaradat andM Ayob ldquoBig Bang-Big Crunch optimizationalgorithm to solve the course timetabling problemrdquo in Proceed-ings of the 10th International Conference on Intelligent SystemsDesign and Applications (ISDA rsquo10) pp 1448ndash1452 December2010

[146] D Debels B De Reyck R Leus and M Vanhoucke ldquoA hybridscatter searchelectromagnetism meta-heuristic for projectschedulingrdquo European Journal of Operational Research vol 169no 2 pp 638ndash653 2006

[147] P Chang S Chen and C Fan ldquoA hybrid electromagnetism-like algorithm for single machine scheduling problemrdquo ExpertSystems with Applications vol 36 no 2 pp 1259ndash1267 2009

[148] A Jamili M A Shafia and R Tavakkoli-Moghaddam ldquoAhybridization of simulated annealing and electromagnetism-like mechanism for a periodic job shop scheduling problemrdquoExpert Systems with Applications vol 38 no 5 pp 5895ndash59012011

[149] M Mirabi S M T Fatemi Ghomi F Jolai and M ZandiehldquoHybrid electromagnetism-like algorithm for the flowshopscheduling with sequence-dependent setup timesrdquo Journal ofApplied Sciences vol 8 no 20 pp 3621ndash3629 2008

[150] B Naderi R Tavakkoli-Moghaddam and M Khalili ldquoElectro-magnetism-like mechanism and simulated annealing algo-rithms for flowshop scheduling problems minimizing the totalweighted tardiness and makespanrdquo Knowledge-Based Systemsvol 23 no 2 pp 77ndash85 2010

[151] H Turabieh S Abdullah and B McCollum ldquoElectromag-netism-like mechanism with force decay rate great deluge forthe course timetabling problemrdquo in Rough Sets and Knowl-edgeTechnology pp 497ndash504 Springer Berlin Germany 2009

[152] C Lee andFChang ldquoFractional-order PID controller optimiza-tion via improved electromagnetism-like algorithmrdquo ExpertSystems with Applications vol 37 no 12 pp 8871ndash8878 2010

[153] S Birbil and O Feyzioglu ldquoA global optimization method forsolving fuzzy relation equationsrdquo in Fuzzy Sets and Systems(IFSA rsquo03) pp 47ndash84 Springer Berlin Germany 2003

[154] P Wu K Yang and Y Hung ldquoThe study of electromagnetism-like mechanism based fuzzy neural network for learning fuzzyif-then rulesrdquo in Knowledge-Based Intelligent Information andEngineering Systems pp 907ndash907 Springer Berlin Germany2005

[155] C Lee C Kuo H Chang J Chien and F Chang ldquoA hybridalgorithm of electromagnetism-like and genetic for recurrentneural fuzzy controller designrdquo in Proceedings of the Interna-tional MultiConference of Engineers and Computer Scientistsvol 1 March 2009


[156] A Yurtkuran and E Emel ldquoA new hybrid electromagnetism-like algorithm for capacitated vehicle routing problemsrdquo ExpertSystems with Applications vol 37 no 4 pp 3427ndash3433 2010

[157] C Tsai H Hung and S Lee ldquoElectromagnetism-like methodbased blind multiuser detection for MC-CDMA interferencesuppression over multipath fading channelrdquo in 2010 Interna-tional Symposium on Computer Communication Control andAutomation (3CA rsquo10) vol 2 pp 470ndash475 May 2010

[158] C-S Tsou and C-H Kao ldquoMulti-objective inventory con-trol using electromagnetism-like meta-heuristicrdquo InternationalJournal of Production Research vol 46 no 14 pp 3859ndash38742008

[159] XWang LGao andC Zhang ldquoElectromagnetism-likemecha-nism based algorithm for neural network trainingrdquo inAdvancedIntelligent Computing Theories and Applications With Aspectsof Artificial Intelligence pp 40ndash45 Springer Berlin Germany2008

[160] Q Wu C Zhang L Gao and X Li ldquoTraining neural networksby electromagnetism-like mechanism algorithm for tourismarrivals forecastingrdquo in Proceedings of the IEEE 5th InternationalConference on Bio-Inspired Computing Theories and Applica-tions (BIC-TA rsquo10) pp 679ndash688 September 2010

[161] P Wu and H Chiang ldquoThe Application of electromagnetism-like mechanism for solving the traveling salesman problemsrdquoinProceeding of the 2005 Chinese Institute of Industrial EngineersAnnual Meeting Taichung Taiwan December 2005

[162] PWu K Yang and H Fang ldquoA revised EM-like algorithm +K-OPTmethod for solving the traveling salesman problemrdquo in 1stInternational Conference on Innovative Computing InformationandControl 2006 (ICICIC rsquo06) vol 1 pp 546ndash549 August 2006

[163] C Su and H Lin ldquoApplying electromagnetism-like mechanismfor feature selectionrdquo Information Sciences vol 181 no 5 pp972ndash986 2011

[164] K C Lee and J Y Jhang ldquoApplication of electromagnetism-likealgorithm to phase-only syntheses of antenna arraysrdquo Progressin Electromagnetics Research vol 83 pp 279ndash291 2008

[165] C Santos M Oliveira V Matos A Maria A C Rocha andL A Costa ldquoCombining central pattern generators with theelectromagnetism-like algorithm for head motion stabilizationduring quadruped robot locomotionrdquo in Proceedings of the2nd International Workshop on Evolutionary and ReinforcementLearning for Autonomous Robot Systems 2009

[166] X Guan X Dai and J Li ldquoRevised electromagnetism-likemechanism for flow path design of unidirectional AGV sys-temsrdquo International Journal of Production Research vol 49 no2 pp 401ndash429 2011


[168] K F Pal ldquoHysteretic optimization for the traveling salesmanproblemrdquo Physica A vol 329 no 1-2 pp 287ndash297 2003

[169] B Goncalves and S Boettcher ldquoHysteretic optimization for spinglassesrdquo Journal of Statistical Mechanics vol 2008 no 1 ArticleID P01003 2008

[170] X Yan andWWu ldquoHysteretic optimization for the capacitatedvehicle routing problemrdquo in Proceedings of the 9th IEEE Interna-tional Conference on Networking Sensing and Control (ICNSCrsquo12) pp 12ndash15 April 2012

[171] J Zha G Zeng and Y Lu ldquoHysteretic optimization for proteinfolding on the latticerdquo in Proceedings of the International Con-ference on Computational Intelligence and Software Engineering(CiSE rsquo10) pp 1ndash4 December 2010

[172] A Kaveh and S Talatahari ldquoA charged system search with afly to boundary method for discrete optimum design of trussstructuresrdquo Asian Journal of Civil Engineering vol 11 no 3 pp277ndash293 2010

[173] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[174] A Kaveh and S Talatahari ldquoCharged system search for optimaldesign of frame structuresrdquoApplied SoftComputing Journal vol12 no 1 pp 382ndash393 2012

[175] A Kaveh and S Talatahari ldquoCharged system search for opti-mum grillage system design using the LRFD-AISC coderdquoJournal of Constructional Steel Research vol 66 no 6 pp 767ndash771 2010

[176] A Kaveh and S Talatahari ldquoGeometry and topology optimiza-tion of geodesic domes using charged system searchrdquo Structuraland Multidisciplinary Optimization vol 43 no 2 pp 215ndash2292011

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


QGA QEA EM QPSO

BB-BC

HQGA

RQGA

QSE

vQEA

IQEA

CFO

BQEA

QICA

HO

VM-APO

EAPO

BGSA

APO

GSA

UBB-CBC

PSOGSA

MOGSA

CQACO

CSS

GIO

GbSA

ECFO

IGOA

QBSO

1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014Timeline

Figure 1 Evolution of physics-inspired optimization algorithms

by quantum mechanics in 1982 paved way for physics-inspired optimization algorithms With this the concept ofquantum computing was developed and in 1995 Narayananand Moore [9] proposed Quantum-Inspired Genetic Algo-rithm (QGA) This is the beginning of physics-inspired opti-mization algorithms After half a decade later in 2002 Hanand Kim proposed Quantum-Inspired Evolutionary Algo-rithm (QEA) In 2004 Quantum-Inspired Particle SwarmOptimization (QPSO) was proposed by Sun et al [10] andin 2007 another swarm-basedQuantum Swarm EvolutionaryAlgorithm (QSE) was proposed by Wang et al [11] Apartfrom the quantummechanics other principles and theoremsof physics also begun to draw the attention of researchersIn 2003 Birbil and Fang [12] proposed Electromagnetism-like (EM) mechanism based on the superposition principleof electromagnetism Big Bang-Big Crunch (BB-BC) [13]based on hypothetical theorem of creation and destructionof the universe was proposed in 2005 Based on Newtonrsquosgravitational law and laws of motion algorithms emergedsuch as CFO [14] by Formato in 2007 GSA by Rashediet al [15] APO by Xie et al [16] in 2009 and GIO byFlores et al [17] in 2011 Hysteretic Optimization (HO) [18]based on demagnetization process was proposed in 2008In 2010 Kaveh and Talatahari proposed CSS [19] based onelectrostatic theorems such as Coulombrsquos law Gaussrsquos law andsuperposition principle from electrostatics andNewtonrsquos lawsof motion In 2011 Shah-Hosseini proposed Spiral Galaxy-Based SearchAlgorithm (GbSA) [20] Jiao et al [21] proposedQICA in 2008 based on quantum theory and immune systemLi et al [22] proposed CQACO based on quantum theoryand ant colony in 2010 Most recently in 2012 Zhang etal [23] proposed IGOA based on gravitational law andimmune system and Jinlong and Gao [24] proposed QBSObased on quantum theory and bacterial forging These majoralgorithms along with their modified improved and hybridversions alongwith the year of proposal are shown in Figure 1We have categorized these algorithms with their variants andtheir notion of inspiration as follows

(A) Newtonrsquos gravitational law(i) Pure physics

(1) CFO(a) Variant (pure physics)(1) ECFO

(2) APO(a) Variant (pure physics)(1) EAPO(2) VM-APO

(3) GSA(a) Variant (pure physics)(1) BGSA(2)MOGSA

(b) Variant (Semiphysics)(1) PSOGSA

(4) GIO(ii) Semiphysics(1) IGOA

(B) Quantum mechanics(i) Pure physics(1) QGA(a) Variant (pure physics)(1) RQGA(2) QGO

(b) Variant (semiphysics)(1)HQGA

(2) QEA(a) Variant (pure physics)(1) BQEA(2) vQEA(3) IQEA

(ii) Semiphysics(1) QPSO(2) QSE(3) QICA(4) CQACO(5) QBSO

(C) Universe theory(i) Pure physics(1) BB-BC(a) Variant (pure physics)(1) UBB-CBC

(2) GbSA





3 Algorithms



119886119901

119905 = 119866

119873

sum119896=1119896 = 119901

119880(119865119896

119905minus 119865119901

119905 ) sdot (119865119896

119905minus 119865119901

119905 )120572

times(119877119896119905minus 119877119901

119905 )

10038171003817100381710038171003817119877119896119905 minus 119877

119901

119905

10038171003817100381710038171003817

120573 (1)



119905and 119865119896

119905are the posi-



119905 ) sdot (119865119896

119905minus 119865119901

119905 )120572




119877119901

119905+1= 119877119901

119905 +1

2119886119901

119905 Δ1199052 (2)




119865 = 11986611989811198982

119903119901 (3)





119865119896

119894119895=

119866119898119894119898119895 (119909119896

119895minus 119909119896119894) if119891 (119883119895) lt 119891 (119883119894)

minus119866119898119894119898119895 (119909119896

119895minus 119909119896119894) if119891 (119883119895) ge 119891 (119883119894)

(5)







119865119896

119894=

119873

sum119895=1

119865119896

119894119895forall119894 = best (6)


V119896119894(119905 + 1) = 119908V119896

119894(119905) +

120582119865119896

119894

119898119894

119909119896

119894(119905 + 1) = 119909

119896

119894(119905) + V119896

119894(119905 + 1)

(7)




119866 (119905) = 119866 (1199050) times (1199050

119905)120573

120573 lt 1 (8)





119872119894 (119905) =119898119894 (119905)

sum119873

119895=1119898119895 (119905)

(9)


follows

119865119889

119894119895= 119866 (119905)

119872119901119894 (119905) times 119872119886119895 (119905)

119877119894119895 (119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

119877119894119895 =10038171003817100381710038171003817119883119894 (119905) 119883119895(119905)

100381710038171003817100381710038172

119865119889

119894(119905) =

119873

sum119895=1119895 = 119894

rand119895119865119889

119894119895(119905)

(10)



119894(119905) is



119886119889

119894(119905) =

119865119889119894

119872119894 (119905) (11)


V119889119894(119905 + 1) = rand119894 times V119889

119894(119905) + 119886

119889

119894(119905)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1)

(12)






119865119894119895 =119872(119891 (119861119894)) sdot 119872 (119891 (119861119895))

10038161003816100381610038161003816119861119894 minus 119861119895

10038161003816100381610038161003816

2119861119894119895 (13)


119872(119891 (119861119894)) = (119891 (119861119894) minus min 119891 (119861)

max119891 (119861) minusmin119891 (119861)


(14)




119865119894 =

119899

sum119895=1

119872(119891 (119861119894)) sdot 119872 (119891 (119861best119895))

10038161003816100381610038161003816119861119894 minus 119861

best119895

10038161003816100381610038161003816

2119861119894119861

best119895 (15)


119881119905+1 = 120594 (119881119905 + 119877 sdot 119862 sdot 119861119896) (16)


119861119896 = radic119872(119891 (119861119894))

10038161003816100381610038161198651198941003816100381610038161003816

119865119894 (17)

120594 =2119896


1003816100381610038161003816100381610038161003816

(18)






119861119905+1 = 119861119905 + 119881119905+1 (20)










|Ψ⟩ = 120572| 0⟩ + 1205731003816100381610038161003816 1⟩ (21)









[12057211205731

10038161003816100381610038161003816100381610038161003816

12057221205732

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120572119898120573119898

] (22)

where |120572119894|2+ |120573119894|

2= 1 119894 = 1 2 119898 With this Q-bit





2+ | cos 120579|2 = 1 For119898Q-



the following

119909119905

119894119895=

1 if random [0 1] gt10038161003816100381610038161003816cos 120579119894119895

10038161003816100381610038161003816

2

0 otherwise(23)


Ψ (119909) =1


(24)


119909 (119905) = 119888 plusmn119871

2ln( 1

119877) (25)








119863119894 = lceil119873119888 times119860 (119902119894)

sum119899

119894=1119860 (119902119894)

rceil (28)



[

[

120572119906

119894

120573119906

119894

]

]

= 119880 (120579119894) times[

[

120572119901

119894

120573119901

119894

]

]

(29)

where120572119906119894and 120573119906


119894and 120573119901

119894are previous



120579119894 = 119896 times 119891 (120572119894 120573119894) (30)




119876119906(119905) = radic1 minus |119876119906 (119905)|

2 (31)





Ψ119894119895 (119898 + 1 119899 119901)

=

radic1 minus (Ψ119894119895 (119898 119899 119901))2

if (120579119894+1119894119895= 0 and 120588 (119898 + 1 119899 119901) lt 1198881)

abs(Ψ119894119895 (119898 119899 119901) times cos 120579119894+1

119894119895


times sin 120579119894+1119894119895

)

otherwise(32)

where 120579119905+1119894119895



120579119894+1

119894119895= 1198901 (119887119895 (119898 119899 119901) minus 119909119894119895 (119898 119899 119901)) (33)


119909119894119895 (119898 + 1 119899 119901)

=

1 if 120574119894119895 (119898 + 1 119899 119901) gt (Ψ119894119895(119898 + 1 119899 119901))2

0 if 120574119894119895 (119898 + 1 119899 119901) le (Ψ119894119895(119898 + 1 119899 119901))2

(34)






119909new119894


119905 (35)










119902119894= 119890(minus119899((119891(119909

119894)minus119891(119909

best))sum119898

119896=1(119891(119909119896)minus119891(119909

best)))) forall119894 (36)




119865119894=

119898

sum119895 = 119894

(119909119895 minus 119909119894)

119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) lt 119891 (119909119894)

(119909119894 minus 119909119895)119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) ge 119891 (119909119894)

forall119894 (37)


119909119894= 119909119894+ 120582

119865119894

10038171003817100381710038171198651198941003817100381710038171003817(RNG)

119894 = 1 2 119898

(38)











119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)


119864119894119895 = 119896119890119902119894

1199032119894119895

(40)


119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)


119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)


119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)


10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)



(44)


119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)




119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)


119875119894119895 =




(48)




Δ119905 (49)


119903new =1



119903new =1

2

119865





119881119895new =119883119895new minus 119883119895old

Δ119905

(52)





itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













3 Algorithms



119886119901

119905 = 119866

119873

sum119896=1119896 = 119901

119880(119865119896

119905minus 119865119901

119905 ) sdot (119865119896

119905minus 119865119901

119905 )120572

times(119877119896119905minus 119877119901

119905 )

10038171003817100381710038171003817119877119896119905 minus 119877

119901

119905

10038171003817100381710038171003817

120573 (1)



119905and 119865119896

119905are the posi-



119905 ) sdot (119865119896

119905minus 119865119901

119905 )120572




119877119901

119905+1= 119877119901

119905 +1

2119886119901

119905 Δ1199052 (2)




119865 = 11986611989811198982

119903119901 (3)





119865119896

119894119895=

119866119898119894119898119895 (119909119896

119895minus 119909119896119894) if119891 (119883119895) lt 119891 (119883119894)

minus119866119898119894119898119895 (119909119896

119895minus 119909119896119894) if119891 (119883119895) ge 119891 (119883119894)

(5)







119865119896

119894=

119873

sum119895=1

119865119896

119894119895forall119894 = best (6)


V119896119894(119905 + 1) = 119908V119896

119894(119905) +

120582119865119896

119894

119898119894

119909119896

119894(119905 + 1) = 119909

119896

119894(119905) + V119896

119894(119905 + 1)

(7)




119866 (119905) = 119866 (1199050) times (1199050

119905)120573

120573 lt 1 (8)





119872119894 (119905) =119898119894 (119905)

sum119873

119895=1119898119895 (119905)

(9)


follows

119865119889

119894119895= 119866 (119905)

119872119901119894 (119905) times 119872119886119895 (119905)

119877119894119895 (119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

119877119894119895 =10038171003817100381710038171003817119883119894 (119905) 119883119895(119905)

100381710038171003817100381710038172

119865119889

119894(119905) =

119873

sum119895=1119895 = 119894

rand119895119865119889

119894119895(119905)

(10)



119894(119905) is



119886119889

119894(119905) =

119865119889119894

119872119894 (119905) (11)


V119889119894(119905 + 1) = rand119894 times V119889

119894(119905) + 119886

119889

119894(119905)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1)

(12)






119865119894119895 =119872(119891 (119861119894)) sdot 119872 (119891 (119861119895))

10038161003816100381610038161003816119861119894 minus 119861119895

10038161003816100381610038161003816

2119861119894119895 (13)


119872(119891 (119861119894)) = (119891 (119861119894) minus min 119891 (119861)

max119891 (119861) minusmin119891 (119861)


(14)




119865119894 =

119899

sum119895=1

119872(119891 (119861119894)) sdot 119872 (119891 (119861best119895))

10038161003816100381610038161003816119861119894 minus 119861

best119895

10038161003816100381610038161003816

2119861119894119861

best119895 (15)


119881119905+1 = 120594 (119881119905 + 119877 sdot 119862 sdot 119861119896) (16)


119861119896 = radic119872(119891 (119861119894))

10038161003816100381610038161198651198941003816100381610038161003816

119865119894 (17)

120594 =2119896


1003816100381610038161003816100381610038161003816

(18)






119861119905+1 = 119861119905 + 119881119905+1 (20)










|Ψ⟩ = 120572| 0⟩ + 1205731003816100381610038161003816 1⟩ (21)









[12057211205731

10038161003816100381610038161003816100381610038161003816

12057221205732

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120572119898120573119898

] (22)

where |120572119894|2+ |120573119894|

2= 1 119894 = 1 2 119898 With this Q-bit





2+ | cos 120579|2 = 1 For119898Q-



the following

119909119905

119894119895=

1 if random [0 1] gt10038161003816100381610038161003816cos 120579119894119895

10038161003816100381610038161003816

2

0 otherwise(23)


Ψ (119909) =1


(24)


119909 (119905) = 119888 plusmn119871

2ln( 1

119877) (25)








119863119894 = lceil119873119888 times119860 (119902119894)

sum119899

119894=1119860 (119902119894)

rceil (28)



[

[

120572119906

119894

120573119906

119894

]

]

= 119880 (120579119894) times[

[

120572119901

119894

120573119901

119894

]

]

(29)

where120572119906119894and 120573119906


119894and 120573119901

119894are previous



120579119894 = 119896 times 119891 (120572119894 120573119894) (30)




119876119906(119905) = radic1 minus |119876119906 (119905)|

2 (31)





Ψ119894119895 (119898 + 1 119899 119901)

=

radic1 minus (Ψ119894119895 (119898 119899 119901))2

if (120579119894+1119894119895= 0 and 120588 (119898 + 1 119899 119901) lt 1198881)

abs(Ψ119894119895 (119898 119899 119901) times cos 120579119894+1

119894119895


times sin 120579119894+1119894119895

)

otherwise(32)

where 120579119905+1119894119895



120579119894+1

119894119895= 1198901 (119887119895 (119898 119899 119901) minus 119909119894119895 (119898 119899 119901)) (33)


119909119894119895 (119898 + 1 119899 119901)

=

1 if 120574119894119895 (119898 + 1 119899 119901) gt (Ψ119894119895(119898 + 1 119899 119901))2

0 if 120574119894119895 (119898 + 1 119899 119901) le (Ψ119894119895(119898 + 1 119899 119901))2

(34)






119909new119894


119905 (35)










119902119894= 119890(minus119899((119891(119909

119894)minus119891(119909

best))sum119898

119896=1(119891(119909119896)minus119891(119909

best)))) forall119894 (36)




119865119894=

119898

sum119895 = 119894

(119909119895 minus 119909119894)

119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) lt 119891 (119909119894)

(119909119894 minus 119909119895)119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) ge 119891 (119909119894)

forall119894 (37)


119909119894= 119909119894+ 120582

119865119894

10038171003817100381710038171198651198941003817100381710038171003817(RNG)

119894 = 1 2 119898

(38)











119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)


119864119894119895 = 119896119890119902119894

1199032119894119895

(40)


119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)


119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)


119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)


10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)



(44)


119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)




119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)


119875119894119895 =




(48)




Δ119905 (49)


119903new =1



119903new =1

2

119865





119881119895new =119883119895new minus 119883119895old

Δ119905

(52)





itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119872119894 (119905) =119898119894 (119905)

sum119873

119895=1119898119895 (119905)

(9)


follows

119865119889

119894119895= 119866 (119905)

119872119901119894 (119905) times 119872119886119895 (119905)

119877119894119895 (119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

119877119894119895 =10038171003817100381710038171003817119883119894 (119905) 119883119895(119905)

100381710038171003817100381710038172

119865119889

119894(119905) =

119873

sum119895=1119895 = 119894

rand119895119865119889

119894119895(119905)

(10)



119894(119905) is



119886119889

119894(119905) =

119865119889119894

119872119894 (119905) (11)


V119889119894(119905 + 1) = rand119894 times V119889

119894(119905) + 119886

119889

119894(119905)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1)

(12)






119865119894119895 =119872(119891 (119861119894)) sdot 119872 (119891 (119861119895))

10038161003816100381610038161003816119861119894 minus 119861119895

10038161003816100381610038161003816

2119861119894119895 (13)


119872(119891 (119861119894)) = (119891 (119861119894) minus min 119891 (119861)

max119891 (119861) minusmin119891 (119861)


(14)




119865119894 =

119899

sum119895=1

119872(119891 (119861119894)) sdot 119872 (119891 (119861best119895))

10038161003816100381610038161003816119861119894 minus 119861

best119895

10038161003816100381610038161003816

2119861119894119861

best119895 (15)


119881119905+1 = 120594 (119881119905 + 119877 sdot 119862 sdot 119861119896) (16)


119861119896 = radic119872(119891 (119861119894))

10038161003816100381610038161198651198941003816100381610038161003816

119865119894 (17)

120594 =2119896


1003816100381610038161003816100381610038161003816

(18)






119861119905+1 = 119861119905 + 119881119905+1 (20)










|Ψ⟩ = 120572| 0⟩ + 1205731003816100381610038161003816 1⟩ (21)









[12057211205731

10038161003816100381610038161003816100381610038161003816

12057221205732

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120572119898120573119898

] (22)

where |120572119894|2+ |120573119894|

2= 1 119894 = 1 2 119898 With this Q-bit





2+ | cos 120579|2 = 1 For119898Q-



the following

119909119905

119894119895=

1 if random [0 1] gt10038161003816100381610038161003816cos 120579119894119895

10038161003816100381610038161003816

2

0 otherwise(23)


Ψ (119909) =1


(24)


119909 (119905) = 119888 plusmn119871

2ln( 1

119877) (25)








119863119894 = lceil119873119888 times119860 (119902119894)

sum119899

119894=1119860 (119902119894)

rceil (28)



[

[

120572119906

119894

120573119906

119894

]

]

= 119880 (120579119894) times[

[

120572119901

119894

120573119901

119894

]

]

(29)

where120572119906119894and 120573119906


119894and 120573119901

119894are previous



120579119894 = 119896 times 119891 (120572119894 120573119894) (30)




119876119906(119905) = radic1 minus |119876119906 (119905)|

2 (31)





Ψ119894119895 (119898 + 1 119899 119901)

=

radic1 minus (Ψ119894119895 (119898 119899 119901))2

if (120579119894+1119894119895= 0 and 120588 (119898 + 1 119899 119901) lt 1198881)

abs(Ψ119894119895 (119898 119899 119901) times cos 120579119894+1

119894119895


times sin 120579119894+1119894119895

)

otherwise(32)

where 120579119905+1119894119895



120579119894+1

119894119895= 1198901 (119887119895 (119898 119899 119901) minus 119909119894119895 (119898 119899 119901)) (33)


119909119894119895 (119898 + 1 119899 119901)

=

1 if 120574119894119895 (119898 + 1 119899 119901) gt (Ψ119894119895(119898 + 1 119899 119901))2

0 if 120574119894119895 (119898 + 1 119899 119901) le (Ψ119894119895(119898 + 1 119899 119901))2

(34)






119909new119894


119905 (35)










119902119894= 119890(minus119899((119891(119909

119894)minus119891(119909

best))sum119898

119896=1(119891(119909119896)minus119891(119909

best)))) forall119894 (36)




119865119894=

119898

sum119895 = 119894

(119909119895 minus 119909119894)

119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) lt 119891 (119909119894)

(119909119894 minus 119909119895)119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) ge 119891 (119909119894)

forall119894 (37)


119909119894= 119909119894+ 120582

119865119894

10038171003817100381710038171198651198941003817100381710038171003817(RNG)

119894 = 1 2 119898

(38)











119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)


119864119894119895 = 119896119890119902119894

1199032119894119895

(40)


119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)


119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)


119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)


10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)



(44)


119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)




119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)


119875119894119895 =




(48)




Δ119905 (49)


119903new =1



119903new =1

2

119865





119881119895new =119883119895new minus 119883119895old

Δ119905

(52)





itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865119894 =

119899

sum119895=1

119872(119891 (119861119894)) sdot 119872 (119891 (119861best119895))

10038161003816100381610038161003816119861119894 minus 119861

best119895

10038161003816100381610038161003816

2119861119894119861

best119895 (15)


119881119905+1 = 120594 (119881119905 + 119877 sdot 119862 sdot 119861119896) (16)


119861119896 = radic119872(119891 (119861119894))

10038161003816100381610038161198651198941003816100381610038161003816

119865119894 (17)

120594 =2119896


1003816100381610038161003816100381610038161003816

(18)






119861119905+1 = 119861119905 + 119881119905+1 (20)










|Ψ⟩ = 120572| 0⟩ + 1205731003816100381610038161003816 1⟩ (21)









[12057211205731

10038161003816100381610038161003816100381610038161003816

12057221205732

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120572119898120573119898

] (22)

where |120572119894|2+ |120573119894|

2= 1 119894 = 1 2 119898 With this Q-bit





2+ | cos 120579|2 = 1 For119898Q-



the following

119909119905

119894119895=

1 if random [0 1] gt10038161003816100381610038161003816cos 120579119894119895

10038161003816100381610038161003816

2

0 otherwise(23)


Ψ (119909) =1


(24)


119909 (119905) = 119888 plusmn119871

2ln( 1

119877) (25)








119863119894 = lceil119873119888 times119860 (119902119894)

sum119899

119894=1119860 (119902119894)

rceil (28)



[

[

120572119906

119894

120573119906

119894

]

]

= 119880 (120579119894) times[

[

120572119901

119894

120573119901

119894

]

]

(29)

where120572119906119894and 120573119906


119894and 120573119901

119894are previous



120579119894 = 119896 times 119891 (120572119894 120573119894) (30)




119876119906(119905) = radic1 minus |119876119906 (119905)|

2 (31)





Ψ119894119895 (119898 + 1 119899 119901)

=

radic1 minus (Ψ119894119895 (119898 119899 119901))2

if (120579119894+1119894119895= 0 and 120588 (119898 + 1 119899 119901) lt 1198881)

abs(Ψ119894119895 (119898 119899 119901) times cos 120579119894+1

119894119895


times sin 120579119894+1119894119895

)

otherwise(32)

where 120579119905+1119894119895



120579119894+1

119894119895= 1198901 (119887119895 (119898 119899 119901) minus 119909119894119895 (119898 119899 119901)) (33)


119909119894119895 (119898 + 1 119899 119901)

=

1 if 120574119894119895 (119898 + 1 119899 119901) gt (Ψ119894119895(119898 + 1 119899 119901))2

0 if 120574119894119895 (119898 + 1 119899 119901) le (Ψ119894119895(119898 + 1 119899 119901))2

(34)






119909new119894


119905 (35)










119902119894= 119890(minus119899((119891(119909

119894)minus119891(119909

best))sum119898

119896=1(119891(119909119896)minus119891(119909

best)))) forall119894 (36)




119865119894=

119898

sum119895 = 119894

(119909119895 minus 119909119894)

119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) lt 119891 (119909119894)

(119909119894 minus 119909119895)119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) ge 119891 (119909119894)

forall119894 (37)


119909119894= 119909119894+ 120582

119865119894

10038171003817100381710038171198651198941003817100381710038171003817(RNG)

119894 = 1 2 119898

(38)











119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)


119864119894119895 = 119896119890119902119894

1199032119894119895

(40)


119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)


119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)


119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)


10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)



(44)


119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)




119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)


119875119894119895 =




(48)




Δ119905 (49)


119903new =1



119903new =1

2

119865





119881119895new =119883119895new minus 119883119895old

Δ119905

(52)





itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













[12057211205731

10038161003816100381610038161003816100381610038161003816

12057221205732

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120572119898120573119898

] (22)

where |120572119894|2+ |120573119894|

2= 1 119894 = 1 2 119898 With this Q-bit





2+ | cos 120579|2 = 1 For119898Q-



the following

119909119905

119894119895=

1 if random [0 1] gt10038161003816100381610038161003816cos 120579119894119895

10038161003816100381610038161003816

2

0 otherwise(23)


Ψ (119909) =1


(24)


119909 (119905) = 119888 plusmn119871

2ln( 1

119877) (25)








119863119894 = lceil119873119888 times119860 (119902119894)

sum119899

119894=1119860 (119902119894)

rceil (28)



[

[

120572119906

119894

120573119906

119894

]

]

= 119880 (120579119894) times[

[

120572119901

119894

120573119901

119894

]

]

(29)

where120572119906119894and 120573119906


119894and 120573119901

119894are previous



120579119894 = 119896 times 119891 (120572119894 120573119894) (30)




119876119906(119905) = radic1 minus |119876119906 (119905)|

2 (31)





Ψ119894119895 (119898 + 1 119899 119901)

=

radic1 minus (Ψ119894119895 (119898 119899 119901))2

if (120579119894+1119894119895= 0 and 120588 (119898 + 1 119899 119901) lt 1198881)

abs(Ψ119894119895 (119898 119899 119901) times cos 120579119894+1

119894119895


times sin 120579119894+1119894119895

)

otherwise(32)

where 120579119905+1119894119895



120579119894+1

119894119895= 1198901 (119887119895 (119898 119899 119901) minus 119909119894119895 (119898 119899 119901)) (33)


119909119894119895 (119898 + 1 119899 119901)

=

1 if 120574119894119895 (119898 + 1 119899 119901) gt (Ψ119894119895(119898 + 1 119899 119901))2

0 if 120574119894119895 (119898 + 1 119899 119901) le (Ψ119894119895(119898 + 1 119899 119901))2

(34)






119909new119894


119905 (35)










119902119894= 119890(minus119899((119891(119909

119894)minus119891(119909

best))sum119898

119896=1(119891(119909119896)minus119891(119909

best)))) forall119894 (36)




119865119894=

119898

sum119895 = 119894

(119909119895 minus 119909119894)

119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) lt 119891 (119909119894)

(119909119894 minus 119909119895)119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) ge 119891 (119909119894)

forall119894 (37)


119909119894= 119909119894+ 120582

119865119894

10038171003817100381710038171198651198941003817100381710038171003817(RNG)

119894 = 1 2 119898

(38)











119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)


119864119894119895 = 119896119890119902119894

1199032119894119895

(40)


119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)


119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)


119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)


10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)



(44)


119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)




119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)


119875119894119895 =




(48)




Δ119905 (49)


119903new =1



119903new =1

2

119865





119881119895new =119883119895new minus 119883119895old

Δ119905

(52)





itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119876119906(119905) = radic1 minus |119876119906 (119905)|

2 (31)





Ψ119894119895 (119898 + 1 119899 119901)

=

radic1 minus (Ψ119894119895 (119898 119899 119901))2

if (120579119894+1119894119895= 0 and 120588 (119898 + 1 119899 119901) lt 1198881)

abs(Ψ119894119895 (119898 119899 119901) times cos 120579119894+1

119894119895


times sin 120579119894+1119894119895

)

otherwise(32)

where 120579119905+1119894119895



120579119894+1

119894119895= 1198901 (119887119895 (119898 119899 119901) minus 119909119894119895 (119898 119899 119901)) (33)


119909119894119895 (119898 + 1 119899 119901)

=

1 if 120574119894119895 (119898 + 1 119899 119901) gt (Ψ119894119895(119898 + 1 119899 119901))2

0 if 120574119894119895 (119898 + 1 119899 119901) le (Ψ119894119895(119898 + 1 119899 119901))2

(34)






119909new119894


119905 (35)










119902119894= 119890(minus119899((119891(119909

119894)minus119891(119909

best))sum119898

119896=1(119891(119909119896)minus119891(119909

best)))) forall119894 (36)




119865119894=

119898

sum119895 = 119894

(119909119895 minus 119909119894)

119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) lt 119891 (119909119894)

(119909119894 minus 119909119895)119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) ge 119891 (119909119894)

forall119894 (37)


119909119894= 119909119894+ 120582

119865119894

10038171003817100381710038171198651198941003817100381710038171003817(RNG)

119894 = 1 2 119898

(38)











119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)


119864119894119895 = 119896119890119902119894

1199032119894119895

(40)


119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)


119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)


119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)


10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)



(44)


119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)




119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)


119875119894119895 =




(48)




Δ119905 (49)


119903new =1



119903new =1

2

119865





119881119895new =119883119895new minus 119883119895old

Δ119905

(52)





itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119902119894= 119890(minus119899((119891(119909

119894)minus119891(119909

best))sum119898

119896=1(119891(119909119896)minus119891(119909

best)))) forall119894 (36)




119865119894=

119898

sum119895 = 119894

(119909119895 minus 119909119894)

119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) lt 119891 (119909119894)

(119909119894 minus 119909119895)119902119894119902119895

1003817100381710038171003817119909119895 minus 119909119894

10038171003817100381710038172

if 119891 (119909119895) ge 119891 (119909119894)

forall119894 (37)


119909119894= 119909119894+ 120582

119865119894

10038171003817100381710038171198651198941003817100381710038171003817(RNG)

119894 = 1 2 119898

(38)











119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)


119864119894119895 = 119896119890119902119894

1199032119894119895

(40)


119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)


119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)


119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)


10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)



(44)


119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)




119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)


119875119894119895 =




(48)




Δ119905 (49)


119903new =1



119903new =1

2

119865





119881119895new =119883119895new minus 119883119895old

Δ119905

(52)





itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119865119894119895 = 119896119890119902119894119902119895

1199032119894119895

(39)


119864119894119895 = 119896119890119902119894

1199032119894119895

(40)


119864119894119895 = 119896119890119902119894

1198863119903119894119895 (41)


119865119894119895 = 119864119894119895119902119895119903119894 minus 11990311989510038171003817100381710038171003817119903119894 minus 119903119895

10038171003817100381710038171003817

(42)


119865119895 = 119896119890119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)


10038171003817100381710038171003817

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(43)



(44)


119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

1003817100381710038171003817100381710038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576 (45)




119865119895 = 119902119895 sum119894119894 = 119895

(119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119875119894119895 (119883119894 minus 119883119895) (47)


119875119894119895 =




(48)




Δ119905 (49)


119903new =1



119903new =1

2

119865





119881119895new =119883119895new minus 119883119895old

Δ119905

(52)





itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












itermax)

119896119886 = 05 (1 +iter

itermax)

(53)




4 Conclusion


Abbreviations








algorithm

References

































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article physics-inspired optimization algorithms...

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