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TestProblemsinOptimizationXin-SheYangDepartmentofEngineering,UniversityofCambridge,CambridgeCB21PZ,UKAbstractTestfunctionsareimportanttovalidatenewoptimizationalgorithmsandtocomparetheperformanceofvariousalgorithms.Therearemanytestfunctionsintheliterature,butthereisnostandardlistorsetoftestfunctionsonehastofollow.Newoptimizationalgorithmsshouldbetestedusingatleastasubsetoffunctionswithdiversepropertiessoastomakesurewhetherornotthetestedalgorithmcansolvecertaintypeofopti-mizationefficiently.Hereweprovideaselectedlistoftestproblemsforunconstrainedoptimization.Citationdetail:X.-S.Yang,Testproblemsinoptimization,in:EngineeringOptimization:AnIntroduc-tionwithMetaheuristicApplications(EdsXin-SheYang),JohnWiley&Sons,(2010).arXiv:1008.0549v1[math.OC]3Aug2010Inordertovalidateanynewoptimizationalgorithm,wehavetovalidateitagainststandardtestfunctionssoastocompareitsperformancewithwell-establishedorexistingalgorithms.Therearemanytestfunctions,sothereisnostandardlistorsetoftestfunctionsonehastofollow.However,varioustestfunctionsdoexist,sonewalgorithmsshouldbetestedusingatleastasubsetoffunctionswithdiversepropertiessoastomakesurewhetherornotthetestedalgorithmcansolvecertaintypeofoptimizationefficiently.Inthisappendix,wewillprovideasubsetofcommonlyusedtestfunctionswithsimpleboundsasconstraints,thoughtheyareoftenlistedasunconstrainedproblemsinliterature.Wewilllistthefunctionformf(x),itssearchdomain,optimalsolutionsx∗and/oroptimalTobjectivevaluef∗.Here,weusex=(x1,...,xn)wherenisthedimension.Ackley’sfunction:nn1v11f(x)=20expux2expcos(2πx)+20+e,(1)−h−5unXii−hnXiiti=1i=1wheren=1,2,...,and32.768x32.768fori=1,2,...,n.Thisfunctionhasthe−≤i≤globalminimumf∗=0atx∗=(0,0,...,0).DeJong’sfunctions:ThesimplestofDeJong’sfunctionsistheso-calledspherefunctionnf(x)=x2,5.12x5.12,(2)Xiii=1−≤≤whoseglobalminimumisobviouslyf∗=0at(0,0,...,0).Thisfunctionisunimodalandconvex.Arelatedfunctionistheso-calledweightedspherefunctionorhyper-ellipsoidfunctionnf(x)=i
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