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1ProblemDefinitionsandEvaluationCriteriafortheCEC2005SpecialSessiononReal-ParameterOptimizationP.N.Suganthan1,N.Hansen2,J.J.Liang1,K.Deb3,Y.-P.Chen4,A.Auger2,S.Tiwari31SchoolofEEE,NanyangTechnologicalUniversity,Singapore,6397982(ETH)Z¨urich,Switzerland3KanpurGeneticAlgorithmsLaboratory(KanGAL),IndianInstituteofTechnology,Kanpur,PIN208016,India4NaturalComputingLaboratory,DepartmentofComputerScience,NationalChiaoTungUniversity,Taiwanepnsugan@ntu.edu.sg,Nikolaus.Hansen@inf.ethz.ch,liangjing@pmail.ntu.edu.sg,deb@iitk.ac.in,ypchen@csie.nctu.edu.tw,Anne.Auger@inf.ethz.ch,tiwaris@iitk.ac.inTechnicalReport,NanyangTechnologicalUniversity,SingaporeAndKanGALReportNumber2005005(KanpurGeneticAlgorithmsLaboratory,IITKanpur)May2005Acknowledgement:WealsoacknowledgethecontributionsbyDrs/ProfessorsMauriceClerc(Maurice.Clerc@WriteMe.com),BogdanFilipic(bogdan.filipic@ijs.si),WilliamHart(wehart@sandia.gov),MarcSchoenauer(Marc.Schoenauer@lri.fr),Hans-PaulSchwefel(hans-paul.schwefel@cs.uni-dortmund.de),AristinPedroBallester(p.ballester@imperial.ac.uk)andDarrellWhitley(whitley@CS.ColoState.EDU).2ProblemDefinitionsandEvaluationCriteriafortheCEC2005SpecialSessiononReal-ParameterOptimizationInthepasttwodecades,differentkindsofoptimizationalgorithmshavebeendesignedandappliedtosolvereal-parameterfunctionoptimizationproblems.Someofthepopularapproachesarereal-parameterEAs,evolutionstrategies(ES),differentialevolution(DE),particleswarmoptimization(PSO),evolutionaryprogramming(EP),classicalmethodssuchasquasi-Newtonmethod(QN),hybridevolutionary-classicalmethods,othernon-evolutionarymethodssuchassimulatedannealing(SA),tabusearch(TS)andothers.Undereachcategory,thereexistmanydifferentmethodsvaryingintheiroperatorsandworkingprinciples,suchascorrelatedESandCMA-ES.Inmostsuchstudies,asubsetofthestandardtestproblems(Sphere,Schwefel's,Rosenbrock's,Rastrigin's,etc.)isconsidered.Althoughsomecomparisonsaremadeinsomeresearchstudies,oftentheyareconfusingandlimitedtothetestproblemsusedinthestudy.Insomeoccasions,thetestproblemandchosenalgorithmarecomplementarytoeachotherandthesamealgorithmmaynotworkinotherproblemsthatwell.Thereisdefinitelyaneedofevaluatingthesemethodsinamoresystematicmannerbyspecifyingacommonterminationcriterion,sizeofproblems,initializationscheme,linkages/rotation,etc.Thereisalsoaneedtoperformascalabilitystudydemonstratinghowtherunningtime/evaluationsincreasewithanincreaseintheproblemsize.Wewouldalsoliketoincludesomerealworldproblemsinourstandardtestsuitewithcodes/executables.Inthisreport,25benchmarkfunctionsaregivenandexperimentsareconductedonsomereal-parameteroptimizationalgorithms.ThecodesinMatlab,CandJavaforthemcouldbefoundat’05TestFunctionszUnimodalFunctions(5):¾F1:ShiftedSphereFunction¾F2:ShiftedSchwefel’sProblem1.2¾F3:ShiftedRotatedHighConditionedEllipticFunction¾F4:ShiftedSchwefel’sProblem1.2withNoiseinFitness¾F5:Schwefel’sProblem2.6withGlobalOptimumonBoundszMultimodalFunctions(20):¾BasicFunctions(7):F6:ShiftedRosenbrock’sFunctionF7:ShiftedRotatedGriewank’sFunctionwithoutBoundsF8:ShiftedRotatedAckley’sFunctionwithGlobalOptimumonBoundsF9:ShiftedRastrigin’sFunctionF10:ShiftedRotatedRastrigin’sFunctionF11:ShiftedRotatedWeierstrassFunctionF12:Schwefel’sProblem2.13¾ExpandedFunctions(2):3F13:ExpandedExtendedGriewank’splusRosenbrock’sFunction(F8F2)F14:ShiftedRotatedExpandedScaffer’sF6¾HybridCompositionFunctions(11):F15:HybridCompositionFunctionF16:RotatedHybridCompositionFunctionF17:RotatedHybridCompositionFunctionwithNoiseinFitnessF18:RotatedHybridCompositionFunctionF19:RotatedHybridCompositionFunctionwithaNarrowBasinfortheGlobalOptimumF20:RotatedHybridCompositionFunctionwiththeGlobalOptimumontheBoundsF21:RotatedHybridCompositionFunctionF22:RotatedHybridCompositionFunctionwithHighConditionNumberMatrixF23:Non-ContinuousRotatedHybridCompositionFunctionF24:RotatedHybridCompositionFunctionF25:RotatedHybridCompositionFunctionwithoutBounds¾Pseudo-RealProblems:Availablefrom~genitor/functions.html.Ifyouhaveanyqueriesontheseproblems,pleasecontactProfessorDarrellWhitley.Email:whitley@CS.ColoState.EDU42.Definitionsofthe25CEC’05TestFunctions2.1UnimodalFunctions:2.1.1.F1:ShiftedSphereFunction2111()_DiiFzfbias==+∑x,=−zxo,12[,,...,]Dxxx=xD:dimensions.12[,,...,]Dooo=o:theshiftedglobaloptimum.Figure2-13-Dmapfor2-DfunctionProperties:¾Unimodal¾Shifted¾Separable¾Scalable¾[100,100]D∈−x,Globaloptimum:*=xo,1(*)1Ff_bias=x=-450AssociatedDatafiles:Name:sphere_func_data.matsphere_func_data.txtVariable:o1*100vectortheshiftedglobaloptimumWhenusing,cuto=o(1:D)Name:fbias_data.matfbias_data.txtVariable:f_bias1*25vector,recordallthe25function’sf_biasi52.1.2.F2:ShiftedSchwefel’sProblem1.222211()()_DijijFzfbias===+∑∑x,=−zxo,12[,,...,]Dxxx=xD:dimensions12[,,...,]Dooo=o:theshiftedglobaloptimumFigure2-23-Dmapfor2-DfunctionProperti