v2007.11.26 - Convex Optimization

252 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.3.3 Optimality of perturbed X ⋆We show that the optimal objective value is unaltered by perturbation (606);id est,i∑〈C , X ⋆ + t j B j 〉 = 〈C , X ⋆ 〉 (617)j=1Proof. From Corollary 4.2.3.0.1 we have the necessary and sufficientrelationship between optimal primal and dual solutions under the assumptionof existence of a relatively interior feasible point:S ⋆ X ⋆ = S ⋆ R 1 R T 1 = X ⋆ S ⋆ = R 1 R T 1 S ⋆ = 0 (618)This means R(R 1 ) ⊆ N(S ⋆ ) and R(S ⋆ ) ⊆ N(R T 1 ). From (607) and (610)we get the sequence:X ⋆ = R 1 R1TX ⋆ + t 1 B 1 = R 2 R2 T = R 1 (I − t 1 ψ(Z 1 )Z 1 )R1TX ⋆ + t 1 B 1 + t 2 B 2 = R 3 R3 T = R 2 (I − t 2 ψ(Z 2 )Z 2 )R2 T = R 1 (I − t 1 ψ(Z 1 )Z 1 )(I − t 2 ψ(Z 2 )Z 2 )R1T.()∑X ⋆ + i i∏t j B j = R 1 (I − t j ψ(Z j )Z j ) R1 T (619)j=1j=1Substituting C = svec −1 (A T y ⋆ ) + S ⋆ from (547),()∑〈C , X ⋆ + i i∏t j B j 〉 =〈svec −1 (A T y ⋆ ) + S ⋆ , R 1 (I − t j ψ(Z j )Z j )j=1〈〉m∑= yk ⋆A ∑k , X ⋆ + i t j B jk=1j=1j=1〈 m〉∑= yk ⋆A k + S ⋆ , X ⋆ = 〈C , X ⋆ 〉 (620)k=1R T 1〉because 〈B i , A j 〉=0 ∀i, j by design (603).