Benchmark-Functions-for-the-CEC’2008-SpecialSessio

1BenchmarkFunctionsfortheCEC’2008SpecialSessionandCompetitiononLargeScaleGlobalOptimizationK.Tang1,X.Yao1,2,P.N.Suganthan3,C.MacNish4,Y.P.Chen5,C.M.Chen5,Z.Yang11NatureInspiredComputationandApplicationsLaboratory(NICAL),DepartmentofComputerScienceandTechnology,theUniversityofScienceandTechnologyofChina,Hefei,Anhui,China2TheCenterofExcellenceforResearchinComputationalIntelligenceandApplications(CERCIA),SchoolofComputerScience,theUniversityofBirminham,Edgbaston,BirminghamB152TT,U.K.3SchoolofEEE,NanyangTechnologicalUniversity,Singapore,6397984SchoolofComputerScience&SoftwareEngineering,theUniversityofWesternAustralia,M002,35StirlingHighway,Crawley,WesternAustralia,60095NaturalComputingLaboratory,DepartmentofComputerScience,NationalChiaoTungUniversity,[e-mail],[e-mail],[e-mail],[e-mail],[e-mail],[e-mail],[e-mail]2Inthepasttwodecades,differentkindsofnature-inspiredoptimizationalgorithmshavebeendesignedandappliedtosolveoptimizationproblems,e.g.,simulatedannealing(SA),evolutionaryalgorithms(EAs),differentialevolution(DE),particleswarmoptimization(PSO),AntColonyOptimisation(ACO),EstimationofDistributionAlgorithms(EDA),etc.Althoughtheseapproacheshaveshownexcellentsearchabilitieswhenapplyingtosome30-100dimensionalproblems,manyofthemsufferfromthecurseofdimensionality,whichimpliesthattheirperformancedeterioratesquicklyasthedimensionalityofsearchspaceincreases.Thereasonsappeartobetwo-fold.First,complexityoftheproblemusuallyincreaseswiththesizeofproblem,andapreviouslysuccessfulsearchstrategymaynolongerbecapableoffindingtheoptimalsolution.Second,thesolutionspaceoftheproblemincreasesexponentiallywiththeproblemsize,andamoreefficientsearchstrategyisrequiredtoexploreallthepromisingregionsinagiventimebudget.Historically,scalingEAstolargesizeproblemshaveattractedmuchinterest,includingboththeoreticalandpracticalstudies.TheearliestpracticalapproachmightbetheparallelismofanexistingEA.Later,cooperativecoevolutionappearstobeanotherpromisingmethod.However,existingworkonthistopicareoftenlimitedtothetestproblemsusedinindividualstudies,andasystematicevaluationplatformisnotavailableintheliteratureforcomparingthescalabilityofdifferentEAs.Inthisreport,6benchmarkfunctionsaregivenbasedon[1]and[2]forhigh-dimensionaloptimization.Allofthemarescalableforanysizeofdimension.ThecodesinMatlabandCforthemareavailableat(Function7-FastFractalDoubleDip)isgeneratedbasedon[3][4].TheCcodeforfunction7hasbeencontributedbyAlesZamudafromtheUniversityofMaribor,Slovenia.ItusestheGJC/CNIinterfacetoruntheJavacodefromC++.Inthepackage,Ccodeisprovidedinaseparatezipfile,named“cec08-f7-cpp.zip”.ThemathematicalformulasandpropertiesofthesefunctionsaredescribedinSection2,andtheevaluationcriteriaaregiveninSection3.1.Summaryofthe7CEC’08TestFunctionszUnimodalFunctions(2):¾F1:ShiftedSphereFunction¾F2:ShiftedSchwefel’sProblem2.21zMultimodalFunctions(5):¾F3:ShiftedRosenbrock’sFunction¾F4:ShiftedRastrigin’sFunction¾F5:ShiftedGriewank’sFunction¾F6:ShiftedAckley’sFunction¾F7:FastFractal“DoubleDip”Function32.Definitionsofthe7CEC’08TestFunctions2.1UnimodalFunctions:2.1.1.F1:ShiftedSphereFunction2111()_DiiFzfbias==+∑x,=−zxo,12[,,...,]Dxxx=xD:dimensions.12[,,...,]Dooo=o:theshiftedglobaloptimum.Figure2-13-Dmapfor2-DfunctionProperties:¾Unimodal¾Shifted¾Separable¾Scalable¾DimensionDas100,500and1000¾[100,100]D∈−x,Globaloptimum:*=xo,1(*)1Ff_bias=x=-450AssociatedDatafiles:Name:sphere__shift_func_data.matVariable:o1*1000vectortheshiftedglobaloptimumWhenusing,cuto=o(1:D)forD=100,50042.1.2.F2:Schwefel’sProblem2.2122()max{||,1}_iiFziDfbias=≤≤+x,=−zxo,12[,,...,]Dxxx=xD:dimensions.12[,,...,]Dooo=o:theshiftedglobaloptimum.Figure2-23-Dmapfor2-DfunctionProperties:¾Unimodal¾Shifted¾Non-separable¾Scalable¾DimensionDas100,500and1000¾[100,100]D∈−x,Globaloptimum:*=xo,1(*)1Ff_bias=x=-450AssociatedDatafiles:Name:schwefel_shift_func_data.matVariable:o1*1000vectortheshiftedglobaloptimumWhenusing,cuto=o(1:D)forD=100,50052.2MultimodalFunctions2.2.1.F3:ShiftedRosenbrock’sFunction12223131()(100()(1))_DiiiiFzzzfbias−+==−+−+∑x,1=−+zxo,12[,,...,]Dxxx=xD:dimensions12[,,...,]Dooo=o:theshiftedglobaloptimum−100−50050100−100−500501000246810x1010Figure2-33-Dmapfor2-DfunctionProperties:¾Multi-modal¾Shifted¾Non-separable¾Scalable¾Havingaverynarrowvalleyfromlocaloptimumtoglobaloptimum¾DimensionDas100,500and1000¾[100,100]D∈−x,Globaloptimum*=xo,*3()3Ff_bias=x=390AssociatedDatafile:Name:rosenbrock_shift_func_data.matVariable:o1*1000vectortheshiftedglobaloptimumWhenusing,cuto=o(1:D)forD=100,50062.2.2.F4:ShiftedRastrigin’sFunction2441()(10cos(2)10)_DiiiFzzfbiasπ==−++∑x,=−zxo,12[,,...,]Dxxx=xD:dimensions12[,,...,]Dooo=o:theshiftedglobaloptimumFigure2-43-Dmapfor2-DfunctionProperties:¾Multi-modal¾Shifted¾Separable¾Scalable¾Localoptima’snumberishuge¾DimensionDas100,500and1000¾[5,5]D∈−x,Globaloptimum*=xo,*4()4Ff_bias=x=-330AssociatedDatafile:Name:rastrigin_shifit_func_data.matVariable:o1*1000ve