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1、OptimizationMethodsandSoftwareVol.22,No.2,April2007,329338AdescentmethodforstructuredmonotonevariationalinequalitiesCAI-HONGYEandXIAO-MINGYUAN*SchoolofManagementScience,HuazhongUniversityofScienceandTechnology,ChinaDepartmentofManagementScienceandEnginee
2、ring,AntaiSchoolofManagement,ShanghaiJiaoTongUniversity,Shanghai,200052,China(Received9March2005;revised29June2005;infinalform22December2005)Thisarticlepresentsadescentmethodforsolvingmonotonevariationalinequalitieswithseparatestructures.Thedescentdirecti
3、onisderivedfromthewell-knownalternatingdirectionsmethod.Theoptimalstepsizealongthedescentdirectionalsoimprovestheefficiencyofthenewmethod.Somenumericalresultsdemonstratethatthenewmethodiseffectiveinpractice.Keywords:Monotonevariationalinequalities;Alterna
4、tingdirectionsmethod;DescentmethodMathematicsSubjectClassification(1991):65D10;65D07;90C251.IntroductionThisarticleisconcernedwiththefollowingmonotonevariationalinequality(VI)withseparatestructures:Findu∈,suchthat(u−u)TF(u)≥0,∀u∈,(1)wherexf(x)u=,F
5、(u)=,={(x,y)
6、x∈X,y∈Y,Ax+By=b},(2)yg(y)X⊂RnandY⊂Rmaregivennonemptyclosedconvexsets;f:X→Rnandg:Y→Rmaregivencontinuousmonotoneoperators;A∈Rr×nandB∈Rr×maregivenmatriceswithfullranksandb∈Rrisagivenvector.Throughoutthisarticleweassumethatr≥mandthatthesolution
7、setof(1)and(2),denotedby∗,isnonempty.TheVI(1)and(2)hasreceivedmuchattentionbecausenumerousapplicationsinoperationsresearch,economics,transportationequilibriumandsooncanbeexplainedbythismodel.Asshowninrefs.[1114],byattachingaLagrangemultipliervectorλ∈Rrt
8、othelinearconstraintsAx+By=b,(1)and(2)canbereformulatedintothefollowingequivalentbutmorecompactform.*Correspondingauthor.Tel.:+1-250-4725690;Fax:+1-250-7218962;Email:xmyuan@hotmail.comOptimizationMethodsandSoftwareISSN1055-6788print/ISSN1029-4937online©2
9、007Taylor&Francishttp://www.tandf.co.uk/journalsDOI:10.1080/10556780600552693330Cai-hongYeandXiao-mingYuanFindw=(x,y,λ)∈Wsuchthat(w−w)TM(w)≥0,∀w∈W,(3)where⎛⎞⎛⎞xf(x)−ATλw=⎝y⎠,M(w)=⎝g(y)−BTλ⎠,W:=X×Y×Rr.(4)λAx+By−bConsequen
