测试函数集

147AdvancedModelingandOptimization,Volume10,Number1,20081AnUnconstrainedOptimizationTestFunctionsCollectionNeculaiAndreiResearchInstituteforInformatics,CenterforAdvancedModelingandOptimization,8-10,AverescuAvenue,Bucharest1,Romania,andAcademyofRomanianScientists.45,SplaiulIndependentei,Bucharest4,RomaniaE-mail:[e-mail]onistogivetotheoptimizationcommunityalargenumberofgeneraltestfunctionstobeusedintestingtheunconstrainedoptimizationalgorithmsandcomparisonsstudies.Foreachfunctionwegiveitsalgebraicexpressionandthestandardinitialpoint.SomeofthetestfnctionsarefromtheCUTEcollectionestablishedbyBongartz,Conn,GouldandToint,[1995],othersarefromMoré,GarbowandHillstrom,[1981],Himmelblau[1972]orfromsomeotherpapersortechnicalreports.1.IntroductionAlwaystheoristsworkinginnonlinearprogrammingarea,aswellaspracticaloptimizersneedtoevaluatenonlinearoptimizationalgorithms.Duetothehypothesisintroducedinordertoprovetheconvergenceandthecomplexityofalgorithms,thetheoryisnotenoughtoestablishtheefficiencyandthereliabilityofamethod.Asaconsequencetheonlywaytoseethe“power”ofanalgorithmremainsitsimplementationincomputercodesanditstestingonlargeclassesoftestproblemsofdifferentstructuresandcharacteristics.Besides,asGeorgeB.Dantzig(1914-2005)said“thefinaltestofatheoryisitscapacitytosolvetheproblemswhichoriginatedit”.Thisisthemainreasonweassembledherethiscollectionoflarge-scaleunconstrainedoptimizationproblemstotestthetheoreticaldevelopmentsinmathematicalprogramming.Nonlinearprogrammingalgorithmsneedtobetestedatleastintwodifferentsenses.Firstly,testingisalwaysprofitableintotheprocessofdevelopmentofanalgorithminordertoevaluatetheideasandthecorrespondingalgebraicprocedures.Clearly,welldesignedtestproblemsareverypowerfulinclarifyingthealgorithmicideasandmechanisms.Secondly,areasonablylargeset1AMO-AdvancedModelingandOptimization:ISSN:1841-4311oftestproblemsmustbeusedinordertogetanideaaboutthehypothesisusedinprovingthequalityofthealgorithm(localandglobalconvergence,complexity)andtocomparealgorithmsatanexperimentallevel.Generally,twotypesof(unconstrained)nonlinearprogrammingproblemscanbeidentified:“artificialproblems”and“real-lifeproblems”.Theartificialnonlinearprogrammingproblemsareusedtoseethebehaviorofthealgorithmsindifferentdifficultsituationslikelongnarrowvalleys,functionswithsignificantnull-spaceeffects,essentiallyunimodalfunctions,functionswithahugenumberofsignificantlocaloptima,etc.Figures1-6presentsometypesofartificialnonlinearfunctioninunconstrainedoptimization.Allofthemareof2variables,thushavingthepossibilityfortheirgraphicalrepresentation.Fig.1.Unimodalfunction.Fig.2.Functionswithsignificantnull-spaceeffects.Fig.3.Essentiallyunimodalfunctions.Fig.4.Functionswithahugenumberofsignificantlocaloptima.Fig.5.Functionswithasmallnumberofsignificantlocaloptima.Fig.6.Functionswhoseglobalstructureprovidesnousefulinformationaboutitsoptima.148149iThemaincharacteristicofartificialnonlinearprogrammingproblemsisthattheyarerelativelyeasytomanipulateandtouseintotheprocessofalgorithmicinvention.Besides,thealgorithmistmayrapidlymodifytheprobleminordertoplacethealgorithmindifferentdifficultconditions.Real-lifeproblems,ontheotherhand,arecomingfromdifferentsourcesofappliedoptimizationproblemslikephysics,chemistry,engineering,biology,economy,oceanography,astronomy,meteorology,etc.Unlikeartificial(unconstrained)nonlinearprogrammingproblems,real-lifeproblemsarenoteasilyavailableandaredifficulttomanipulate.Theymayhavecomplicatedalgebraic(ordifferential)expressions,maydependonahugeamountofdata,andpossiblearedependentonsomeparameterswhichmustbeestimatedinaspecificway.Averynicecollectionofreal-lifeunconstrainedoptimizationproblemsisthatgivenbyAverick,etal.[1991,1992].Inthiscollectionweconsideronlyartificialunconstrainedoptimizationtestproblems.Allofthemarepresentedinextendedorgeneralizedform.ThemaindifferencebetweentheseformsisthatwhiletheproblemsingeneralizedformhavetheHessianmatrixasablockdiagonalmatrix,theextendedformshavetheHessianasamulti-diagonalmatrix.Manyindividualshavecontributed,eachoftheminimportantways,tothepreparationofthiscollection.Wedonotmentionthemhere.AnimportantsourceofproblemswastheCUTEcollectionestablishedbyBongartz,Conn,GouldandToint,[1995].SomeotherproblemsarefromMoré,GarbowandHillstrom,[1981],Himmelblau[1972]orareextractedfromsomeotherpapersortechnicalreports.Generally,theproblemsinextendedformsareslightlymoredifficulttobesolved.2.UnconstrainedOptimizationTestFunctionsExtendedFreudenstein&Rothfunction:()fxxxxxiiiin()(())/=−++−−−=∑135221222212()+−+++−−29114212222xxxxiii(())i,x0052052052=−−−[.,,.,,,.,].…ExtendedTrigonometricfunction:fxnxixxjjniiin()cos(cos)sin,=−⎛⎝⎜⎞⎠⎟+−−⎛⎝⎜⎞⎠⎟==∑∑1211x0020202=[.,.,,.].…ExtendedRosenbrockfunction:()()fxcxxxiiiin(),/=−+−−−=∑22122212121x0121121=−−[.,,,.,].…c=100.GeneralizedRosenbrockfunction:()()fxcxxxiiiin(),=−+−+=−∑1222111x0121121=−−[.,.,.,],…fxcxxxiiiin(),/=−+−−−=∑22132212121c=100.ExtendedWhite&Holstfunctio