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1、IntroductiontoSemidefiniteProgramming(SDP)RobertM.Freund1IntroductionSemidefiniteprogramming(SDP)isthemostexcitingdevelopmentinmathematicalprogramminginthe1990’s.SDPhasapplicationsinsuchdiversefieldsastraditionalconvexconstrainedoptimization,controltheory,and
2、combinatorialoptimization.BecauseSDPissolvableviainteriorpointmethods,mostoftheseapplicationscanusuallybesolvedveryefficientlyinpracticeaswellasintheory.2ReviewofLinearProgrammingConsiderthelinearprogrammingprobleminstandardform:LP:minimizec·xs.t.ai·x=bi,i=1
3、,...,mx∈�n+.Herexisavectorofnvariables,andwewrite“c·x”fortheinner-product�n“j=1cjxj”,etc.Also,�n+:={x∈�n
4、x≥0},andwecall�n+thenonnegativeorthant.Infact,�n+isaclosedconvexcone,whereKiscalledaclosedaconvexconeifKsatisfiesthefollowingtwoconditions:1•Ifx,w∈K,the
5、nαx+βw∈Kforallnonnegativescalarsαandβ.•Kisaclosedset.Inwords,LPisthefollowingproblem:“Minimizethelinearfunctionc·x,subjecttotheconditionthatxmustsolvemgivenequationsai·x=bi,i=1,...,m,andthatxmustlieintheclosedconvexconeK=�n+.”Wewillwritethestandardlinearpr
6、ogrammingdualproblemas:�mLD:maximizeyibii=1�ms.t.yiai+s=ci=1s∈�n+.GivenafeasiblesolutionxofLPandafeasiblesolution(y,s)ofLD,the�m�mdualitygapissimplyc·x−i=1yibi=(c−i=1yiai)·x=s·x≥0,becausex≥0ands≥0.WeknowfromLPdualitytheorythatsolongastheprimalproblemLPisfe
7、asibleandhasboundedoptimalobjectivevalue,thentheprimalandthedualbothattaintheiroptimawithnodualitygap.Thatis,thereexistsx∗and(y∗,s∗)feasiblefortheprimalanddual,respectively,�suchthatc·x∗−mi=1yi∗bi=s∗·x∗=0.3FactsaboutMatricesandtheSemidefiniteConeIfXisann×nm
8、atrix,thenXisapositivesemidefinite(psd)matrixifvTXv≥0foranyv∈�n.2IfXisann×nmatrix,thenXisapositivedefinite(pd)matrixifvTXv>0foranyv∈�n,v=�0.LetSndenotethesetofsymmetricn×nmatrices,andletS+ndenotethesetofpositivesemidefinite(psd)n×nsymmetricmatrices.Similarlyl
9、etS+n+denotethesetofpositivedefinite(pd)n×nsymmetricmatrices.LetXandYbeanysymmetricmatrices.Wewrite“X�0”todenotethatXissymmetricandpositivesemidefinite,andwewrite“X�Y”todenotethatX−Y�0.Wewrite“X�0”todenotethatX
