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1、OperationsResearchLettersOperationsResearchLetters34(2006)481–490www.elsevier.com/locate/orlSolvingasymmetricvariationalinequalitiesviaconvexoptimizationMicheleAghassia,DimitrisBertsimasb,?,GeorgiaPerakisbaOperationsResearchCenter,MassachusettsInstituteofTechnology,E40-131,Cambrid
2、ge,MA02139,USAbSloanSchoolofManagementandOperationsResearchCenter,MassachusettsInstituteofTechnology,E53-363,Cambridge,MA02139,USAReceived20May2005;accepted13September2005Availableonline6December2005AbstractUsingduality,wereformulatetheasymmetricvariationalinequality(VI)problemove
3、raconicregionasanoptimizationproblem.Wegivesuf?cientconditionsfortheconvexityofthisreformulation.WetherebyidentifyaclassofVIsthatincludesmonotoneaf?neVIsoverpolyhedra,whichmaybesolvedbycommercialoptimizationsolvers.?2005ElsevierB.V.Allrightsreserved.Keywords:Variationalinequalitie
4、s;Convexoptimization;Duality1.Introductionofsystemsofequations,complementarityproblems,andaclassof?xedpointproblems.Inaddition,foranyThevariationalinequality(VI)problemhasengagedoptimizationproblemoveraclosed,convexfeasiblemembersoftheoptimization,mathematics,transporta-region,the
5、?rst-orderoptimalityconditionscomprisetionscience,engineering,andeconomicscommuni-aVI.Accordingly,theVIproblemalsogeneralizesties.GivenasetK?RnandamappingF:K→convexoptimization.Rn,theVIproblem,denotedVI(K,F),isto?ndanForacompletediscussionandhistoryoftheVIx?∈Ksuchthatproblemandass
6、ociatedsolutionmethods,wereferthe??interestedreadertotherecentsurveytextbyFacchineiF(x)(x?x)0?x∈K.(1)andPang[12]andthemonographbyPatriksson[24].VIs,?rstintroducedbyStampacchiaandhiscollab-ThesurveyarticlebyHarkerandPang[16]andtheorators[18,19,23,27,28],subsumemanyotherwell-Ph.D.
7、thesisofHammond[15],aswellastherefer-studiedmathematicalproblems,includingthesolutionencestherein,alsoprovideinsightfulreviewsoftheVIproblemandassociatedalgorithms.?OneclassoftechniquesforsolvingtheVIproblemCorrespondingauthor.E-mailaddresses:maghassi@alum.mit.edu(M.Aghassi),explo
8、itsthefactthattheKarush–Kuhn–Tuck
