v2009.01.01 - Convex Optimization

260 CHAPTER 4. SEMIDEFINITE PROGRAMMING
4.2.3.0.1 Corollary. Optimality and strong duality. [312,3.1]
[342,1.3.8] For semidefinite programs (584P) and (584D), assume primal
and dual feasible sets A ∩ S n + ⊂ S n and C ∗ ⊂ S n × R m (596) are nonempty.
ThenX ⋆ is optimal for (P)
S ⋆ , y ⋆ are optimal for (D)
the duality gap 〈C,X ⋆ 〉−〈b, y ⋆ 〉 is 0
if and only if
i) ∃X ∈ A ∩ int S n + or ∃S , y ∈ rel int C ∗
ii) 〈S ⋆ , X ⋆ 〉 = 0
and
⋄
For symmetric positive semidefinite matrices, requirement ii is equivalent
to the complementarity (A.7.4)
〈S ⋆ , X ⋆ 〉 = 0 ⇔ S ⋆ X ⋆ = X ⋆ S ⋆ = 0 (609)
Commutativity of diagonalizable matrices is a necessary and sufficient
condition [176,1.3.12] for these two optimal symmetric matrices to be
simultaneously diagonalizable. Therefore
rankX ⋆ + rankS ⋆ ≤ n (610)
Proof. To see that, the product of symmetric optimal matrices
X ⋆ , S ⋆ ∈ S n must itself be symmetric because of commutativity. (1378) The
symmetric product has diagonalization [10, cor.2.11]
S ⋆ X ⋆ = X ⋆ S ⋆ = QΛ S ⋆Λ X ⋆Q T = 0 ⇔ Λ X ⋆Λ S ⋆ = 0 (611)
where Q is an orthogonal matrix. The product of the nonnegative diagonal Λ
matrices can be 0 if their main diagonal zeros are complementary or coincide.
Due only to symmetry, rankX ⋆ = rank Λ X ⋆ and rankS ⋆ = rank Λ S ⋆ for
these optimal primal and dual solutions. (1363) So, because of the
complementarity, the total number of nonzero diagonal entries from both Λ
cannot exceed n .