非光滑非线性互补问题地牛顿法

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1、20116152June,2011OperationsResearchTransactionsVol.15No.2ANewtonMethodforaNonsmoothNonlinearComplementarityProblem∗GAOYan1AbstractThispaperisdevotedtoanonlinearcomplementarityproblemwithnonsmoothdata.Thenonlinearcomplementarityproblemisreformulatedasasystemofnon

2、smoothequations.Then,aNewtonmethodforsolvingthenonsmoothequationsisproposed.IneachiterationoftheNewtonmethod,anelementoftheB-differentialofrelatedfunctions,notnonlinearcomplementarityfunction,isrequired.Thesuperlinearconvergenceisshown.Keywordsnonlinearcomplementarityprob

3、lem,nonsmoothanalysis,optimization,NewtonmethodsChineseLibraryClassificationO222010MathematicsSubjectClassification90C30,65H10.)'%1/0,+!#-($"O22&*$"90C30,65H100IntroductionThenonlinearcomplementarityproblemF(x)>0,x>0,xTF(x)=0,(0.1)whereF:ℜn→ℜnandx∈ℜnistofindasolutionx∈ℜn,w

4、hichsatisfies(0.1).Thecomplementarityproblemplaysanimportantroleineconomicsequilibrium,systemengineering,optimizationandothers.IthasbeenstudiedextensivelywhenFissmooth,see2009Æ16
∗SupportedbytheNationalScienceFoundationofChina(undergrant:10671126),ShanghaiMunicipalC

5、ommitteeofScienceandTechnology(undergrant:10550500800)andShanghaiLeadingDisciplineProject(undergrant:S30501).1.SchoolofManagementUniversityofShanghaiforScienceandTechnology,Shanghai200093,China;7100722GAOYan15forinstance[3-4,6]andreferencestherein.Basedonn

6、olinearcomplementarityfunction,anonlinearcomplementarityproblemisequivalentlytransformedintoasystemofnonsmoothequations.AgeneralizedNewtonmethodisusedtosolvethesystemofnonsmoothequations.ForthecasewhereFisnonsmoothfunction,toourknowledge,only[1-2,5]dealtwiththeproblem(0.

7、1).Theytransform(0.1)intoaunconstrainedoptimization,thensolveitbynonsmoothoptimizationmethod.Inthepresentpaper,wetrytostudynonlinearcomplementaritywithnonsmoothdata.Wefirstreformulatethenonlinearcomplementarityproblemasasystemofnonsmoothequation,thenproposeaNewtontosolvet

8、henonsmoothequations.LetusconsiderthefollowingnonlinearcomplementarityproblemF(x)>0,Z(x)>0,Z(x)TF(x)=0,