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1、8.2 An ExampleQingna Li(BIT)8.2 An Example1/8ExampleConsider the following optimization problem with equalityand inequality constraintsminxIRnf(x)s.t.ci(x)=0,i E:=1,.,m,hi(x)0,i I:=1,.,k.(1.1)Qingna Li(BIT)8.2 An Example2/8LetG(x)=c1(x).cm(x)h1(x).hk(x),K=0m IRk+.Observed thatK=IRm IRk+.Let=v IRm+k,
2、IRm,v IRk.Qingna Li(BIT)8.2 An Example3/8Recall KKT conditionxL(x,)=0,G(x)K,G(x)=0,K,So we have K IRm,v IRk+.G(x)K ci(x)=0,i E,hi(x)0,i I.u,G(x)=0 iEici(x)+iIvihi(x)=0iIvihi(x)=0(ci(x)=0)vihi(x)=0,i I(hi(x)0,vi 0).Qingna Li(BIT)8.2 An Example4/8The KKT condition of(1.1)can be written asf(x)iEici(x)i
3、Ivihi(x)=0,ci(x)=0,i E,hi(x)0,vi 0,vihi(x)=0 i I.Qingna Li(BIT)8.2 An Example5/8ExampleConsidering linear programmingminxIRnc,xs.t.Ax=b,x 0,(1.2)where A IRmn,b IRmare given.Qingna Li(BIT)8.2 An Example6/8The Lagrange functionL(x,)=c,x ,Ax b ,x(1.3)where IRm,IRn.The KKT condition of(1.2)can be written asxL(x,)=c TA =0,Ax b=0,x 0,0,x=0.Qingna Li(BIT)8.2 An Example7/8SummaryAn exampleAbstract form Vector formQingna Li(BIT)8.2 An Example8/8