v2007.09.13 - Convex Optimization

240 CHAPTER 4. SEMIDEFINITE PROGRAMMINGFor symmetric positive semidefinite matrices, requirement ii is equivalentto the complementarity (A.7.4)〈S ⋆ , X ⋆ 〉 = 0 ⇔ S ⋆ X ⋆ = X ⋆ S ⋆ = 0 (571)Commutativity of diagonalizable matrices is a necessary and sufficientcondition [149,1.3.12] for these two optimal symmetric matrices to besimultaneously diagonalizable. ThereforerankX ⋆ + rankS ⋆ ≤ n (572)Proof. To see that, the product of symmetric optimal matricesX ⋆ , S ⋆ ∈ S n must itself be symmetric because of commutativity. (1275) Thesymmetric product has diagonalization [9, cor.2.11]S ⋆ X ⋆ = X ⋆ S ⋆ = QΛ S ⋆Λ X ⋆Q T = 0 ⇔ Λ X ⋆Λ S ⋆ = 0 (573)where Q is an orthogonal matrix. The product of the nonnegative diagonal Λmatrices 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 ⋆ forthese optimal primal and dual solutions. (1260) So, because of thecomplementarity, the total number of nonzero diagonal entries from both Λcannot exceed n .When equality is attained in (572)rankX ⋆ + rankS ⋆ = n (574)there are no coinciding main diagonal zeros in Λ X ⋆Λ S ⋆ , and so we have whatis called strict complementarity. 4.10 Logically it follows that a necessary andsufficient condition for strict complementarity of an optimal primal and dualsolution isX ⋆ + S ⋆ ≻ 0 (575)The beauty of Corollary 4.2.3.0.1 is its conjugacy; id est, one can solveeither the primal or dual problem and then find a solution to the other via theoptimality conditions. When a dual optimal solution is known, for example,a primal optimal solution belongs to the hyperplane {X | 〈S ⋆ , X〉=0}.4.10 distinct from maximal complementarity (4.1.1).