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1、ComputOptimAppl(2012)52:439461DOI10.1007/s10589-011-9417-zAninexactparallelsplittingaugmentedLagrangianmethodformonotonevariationalinequalitieswithseparablestructuresMinTao·XiaomingYuanReceived:23July2009/Publishedonline:3June2011©SpringerScience+BusinessMedia,LLC2011AbstractSplittingmetho
2、dshavebeenextensivelystudiedinthecontextofconvexprogrammingandvariationalinequalitieswithseparablestructures.Recently,apar-allelsplittingmethodbasedontheaugmentedLagrangianmethod(abbreviatedasPSALM)wasproposedinHe(Comput.Optim.Appl.42:195212,2009)forsolvingvariationalinequalitieswithsepara
3、blestructures.Inthispaper,weproposetheinexactversionofthePSALMapproach,whichsolvestheresultingsubproblemsofPSALMapproximatelybyaninexactproximalpointmethod.FortheinexactPSALM,theresultingproximalsubproblemshaveclosed-formsolutionswhentheproximalpa-rametersandinexacttermsarechosenappropriat
4、ely.WeshowtheefficiencyoftheinexactPSALMnumericallybysomepreliminarynumericalexperiments.KeywordsVariationalinequalities·Splittingmethod·Parallelmethod·Proximalpointmethod·AugmentedLagrangianmethod·Prediction-correctionmethod1IntroductionLetΩ⊂RnbeanonemptyclosedconvexsetandFbeacontinuousmap
5、pingfromRnintoitself.Thevariationalinequality(VI)problem,denotedbyVI(Ω,F),istofindu∈Ωsuchthat(u−u)TF(u)≥0,∀u∈Ω,(1.1)M.Tao()SchoolofScience,NanjingUniversityofPostsandTelecommunications,#9CultureGardensRoad,Nanjing210046,Jiangsu,Chinae-mail:taom@njupt.edu.cnX.YuanDepartmentofMathematics,H
6、ongKongBaptistUniversity,HongKong,Chinae-mail:xmyuan@hkbu.edu.hk440M.Tao,X.YuanwhereTdenotesthestandardinnerproduct.Inthispaper,weconsidertheVI(Ω,F)withthefollowingseparablestructure:xf(x)u=,F(u)=,(1.2)yg(y)andΩ:={(x,y)
7、Ax+By=b,x∈X,y∈Y},(1.3)whereX⊂Rn1andY⊂Rn2arenonemptyclosedandconvex
8、sets;A∈Rm×n1andB∈Rm×n2aregivenmatrices;f:X→Rn1andg:Y→Rn2aregivenmono-tonemappings;b∈Rmisagivenvectorandn1+n2=n.ForwideapplicationsofVI(Ω,F)withtheseparablestructure(1.2)(1.3),seee.g.[2,11,19].Thefavorableseparablestructureof(1.2)(1.3)hasinspiredmanysplittingty
