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1、___________________________________________________________________________http://www.paper.edu.cnRobustSemi-definiteProgrammingAndItsApproximationSunChuren∗InstituteofEconomicsandTrade,201600ShanghaiInstituteofForeignTrade,Songjiang,Shanghai,ChinaE-mail:sunchuren@sohu.comHuangLei†DepartmentofTe
2、chnology,BridgeMillofHami,Xinjiang,ChinaE-mail:valleyh@sohu.comAbstractResearchinrobustoptimizationishotatpresent.InthisarticleweobserverobustTPnsemi-definiteprogrammingoftheformmin{cx
3、∀s∈U,A0(s)+i=1xiAi(s)0}whereUisaunitball,x,c∈RnandAi(x),i=0,···,maredatamatrices.ThisproblemisgenerallyNP-hard
4、anditsdeterministicallyequivalentproblemisalsodifficulttobefound.Inthispaper,weshowthatwhenthedatamatricesare2×2and3×3-dimensional,wecantransfertheproblemintoanoptimizationproblemwithquadraticmatrixinequalityconstraints,basedonanotherequivalentrobustoptimizationproblemconvertedfromtheoriginalone.
5、Also,wegiveanotherproofthatrobustsemi-definiteprogrammingproblemisgenerallyNP-hard.Forpracticalcomputation,weposeamethodtoapproximativelysolvetheoriginalrobustsemi-definiteprogrammingproblem.Keywords.Robustsemi-definiteprogramming;Co-NP-hard;Quadraticmatrixinequality;Polyhedralapproximation;S-lemm
6、a;AMSsubjectclassification.90C34;90C901IntroductionRecently,semi-definiteprogrammingbecomesamain-streamresearchdirectioninop-erationsresearchbecauseofthearisingofthepowerfulinteriorpointmethoddevelopedfirstbyKarmarkar[1],whichwaslaterextendedtomoregeneralconvexconicoptimiza-tionproblemsbyNesterova
7、ndNemirovski[2].Duringthepasttenyearsofresearchin∗SunChuren,Ph.DcandidateofCUHK;Postaddress:TheInstituteofEconomicsandTrade,TheInstituteofForeignTradeofShanghai,Songjiang,Shanghai,201600;Email:sunchuren@sohu.com.†HuangLei,Female,DepartmentofTechnology,BridgeMillofHami,Xinjiang;Email:val-leyh@so
8、hu.com.1thisfield,relativelycompletetheoryhasbeenestablished,efficientalgorithmshavebeendesignedandpowerfulsolvershavebeensetup.Also,manyapplicationshavebeenfoundinvastvarietyoffields,suchasrobuststatistics,portfolioselection,trusstop
