v2009.01.01 - Convex Optimization

4.3. RANK REDUCTION 273
4.3.3 Optimality of perturbed X ⋆
We show that the optimal objective value is unaltered by perturbation (643);
id est,
i∑
〈C , X ⋆ + t j B j 〉 = 〈C , X ⋆ 〉 (654)
j=1
Proof. From Corollary 4.2.3.0.1 we have the necessary and sufficient
relationship between optimal primal and dual solutions under the assumption
of existence of a relatively interior feasible point:
S ⋆ X ⋆ = S ⋆ R 1 R T 1 = X ⋆ S ⋆ = R 1 R T 1 S ⋆ = 0 (655)
This means R(R 1 ) ⊆ N(S ⋆ ) and R(S ⋆ ) ⊆ N(R T 1 ). From (644) and (647)
we get the sequence:
X ⋆ = R 1 R1
T
X ⋆ + t 1 B 1 = R 2 R2 T = R 1 (I − t 1 ψ(Z 1 )Z 1 )R1
T
X ⋆ + 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 )R1
T
.
(
)
∑
X ⋆ + i i∏
t j B j = R 1 (I − t j ψ(Z j )Z j ) R1 T (656)
j=1
j=1
Substituting C = svec −1 (A T y ⋆ ) + S ⋆ from (584),
(
)
∑
〈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 j
k=1
j=1
j=1
〈 m
〉
∑
= yk ⋆A k + S ⋆ , X ⋆ = 〈C , X ⋆ 〉 (657)
k=1
R T 1
〉
because 〈B i , A j 〉=0 ∀i, j by design (640).