1、8.3 Dual ProblemQingna Li(BIT)8.3 Dual Problem 1/13Dual ProblemConsidering p roblemminxXf(x)s.t.G(x)K.(1.1)its dual p roblem issup K infxXL(x,):=sup K(),(1.2)where()=infxXL(x,),L(x,)=f(x)G(x),.Qingna Li(BIT)8.3 Dual Problem 2/13ExampleThe dual p roblem of quadratic p rogramming.Consideringthe quadra
2、tic p rogrammingminx I Rn12xTQx+c,x s.t.Ax=b,Bx d 0.(1.3)where Q is symmetric p ositive semi-definite,A I Rp n,b I Rp,B I Rq n,d I Rqa re given.Qingna Li(BIT)8.3 Dual Problem 3/13G(x)=Ax bBx d,K=0 p I Rq+.Notice thatK=I Rp I Rq+.Let=v I Rp+q,I Rp,v I Rq.So w e have K I Rp,v I Rq+.G(x)K Ax b=0,Bx d 0
3、.Qingna Li(BIT)8.3 Dual Problem 4/13The Lagrange function isL(x,)=12xTQx+c,x G(x),=12xTQx+c,x Ax bBx d,v=12xTQx+c,x Ax b,Bx d,v=12xTQx+x,c AT BTv+b,+d,v.Qingna Li(BIT)8.3 Dual Problem 5/13Notice that Q is symmetric p o sitive semi-definite,thep robleminfx I RnL(x,)(1.4)admits a unique minimum p oint
4、,and xL(x,)=0(1.5)is the suicient and necessa ry optimalit y condition to findthe minimizer.Qingna Li(BIT)8.3 Dual Problem 6/13(1.5)is equivalent toQx+c AT BTv=0.Equivalently,x=Q 1(c+AT+BTv)is the minimizer of(1.4).infx I RnL(x,)=L(x,)=12(c+AT+BTv)TQ 1(c+AT+BTv)+Q 1(c+AT+BTv),c AT v+,b+v,d=12(c+AT+B
5、Tv)Q 1(c+AT+BTv)+,b+v,d Qingna Li(BIT)8.3 Dual Problem 7/13The dual p roblem issup K infxXL(x,)=sup K()=sup I Rp,v I Rq+(,v).i.e.max I Rp,v I Rq(,v)s.t.v 0.where(,v)=12(c+AT+BTv)Q 1(c+AT+BTv)+,b+v,d.Qingna Li(BIT)8.3 Dual Problem 8/13A QuestionIf Q 0 but Q is not p ositive definite,ho w to deriv eth
6、e dual p roblem?Ho w to solve it ei ciently?Restricted W olfe dual T echnique and SymmetricGaussian-Siedel Decomp ositionb y Prof.Defeng Suns resea rch groupXudong L i,Defeng Sun,and Kim Chuan T oh,QSDPNAL:A t w o-phase augmented Lagrangianmetho d fo r co nvex quadratic semidefinitep rogramming,Math
7、ematical ProgrammingComputation,10(2018)703-743.Qingna Li(BIT)8.3 Dual Problem 9/13Indicato r function and Supp o rt F unctionIndicato r f unction of set C:(x|C)=0,x C,+,otherwise.(1.6)min f(x)s.t.x C min f(x)+(x|C)(1.7)Qingna Li(BIT)8.3 Dual Problem 10/13Conjugate f unction of f at x:f(x)=supx x,x
8、f(x).(1.8)The conjuga te function of(|C)at x is referred asthe supp o rt fun ction of C at x,denoted as(x|C).The conjugate function is an imp o rtant to ol to derive dualp roblems fo r mo re complicated p roblems.Qingna Li(BIT)8.3 Dual Problem 11/13Summa ryAbstract fo rm o f dual p roblemMo re to ols in dualit y theo ryQingna Li(BIT)8.3 Dual Problem 12/13Summa ry of Chapter 8KKT conditi onDual p roblemMo re questions a nd to olsQingna Li(BIT)8.3 Dual Problem 13/13
