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1、LineSearchAlgorithmswithGuaranteedSufficientDecreaseJORGEJ.MOREandDAVIDJ.THUENTEArgonneNationalLaboratoryThedevelopmentofsoftwareforminimizationproblemsisoftenbasedonalinesearchmethod.Weconsiderlinesearchmethodsthatsatisfysufficientdecreaseandcurvatureconditions,andformulatetheproblemofdeter
2、miningapointthatsatisfiesthesetwoconditionsintermsoffindingapointinasetT(p),Wedescribeasearchalgorithmforthisproblemthatproducesasequenceofiteratesthatconvergetoapointin2’(p-)andthat,exceptforpathologicalcases,terminatesinafinitenumberofsteps.Numericalresultsforanimplementationofthesearchalg
3、orithmonasetoftestfunctionsshowthatthealgorithmterminateswithinasmallnumberofiterations.CategoriesandSubjectDescriptorsG.16[NumericalAnalysis]:Optimization—corzstralneclopttmtzatton;gradlentmethods;nonlinearprogramming:G.4[MathematicsofComputing]:MathematicalSoftware—algorLthmanalysls;effici
4、ency;reliabilityandrobustnessGeneralTerms:AlgorithmsAdditionalKeyWordsandPhrases:Conjugategradientalgorithms,linesearchalgorithms,nonlinearoptimization,truncatedNewtonalgorithms,variablemetricalgorithms1.INTRODUCTIONGivenacontinuouslydifferentiablefunction@:R~Rdefinedon[0,CO)with+’(0)<0,andc
5、onstantspandqin(O,1),weareinterestedinfindingana>0suchthato(a)<$NO)+~$b’(o)a(1.1)and14’(CI)I<7710’(0)1.(1.2)Thedevelopmentofasearchprocedurethatsatisfiestheseconditionsisacrucialingredientinalinesearchmethodforminimization.Thesearchalgorithmdescribedinthispaperhasbeenusedbyseveralauthors,for
6、ThisworkwassupportedbytheOfficeofScientificComputing,U.S.DepartmentofEnergy,undercontractW-31-109-Eng-38Authors’addresses:J.J.Mor6,MathematicsandComputerScienceDivision,ArgonneNationalLaboratory,9700SouthCassAvenue,Argonne,IL60439;D.JThuente,DepartmentofMathe-maticalSciences,Indiana-PurdueUn
7、iversity,FortWayne,IN46805,Permissiontocopywithoutfeeallorpartofthismaterialisgrantedprovidedthatthecopiesarenotmadeordistributedfordirectcommercialadvantage,theACMcopyrightnoticeandthetitleofthepublicationanditsdateappear,andnoticeisgiventhatcopyi
