v2007.09.13 - Convex Optimization

248 CHAPTER 4. SEMIDEFINITE PROGRAMMINGwhose rank does not satisfy upper bound (232), we posit existence of a setof perturbations{t j B j | t j ∈ R , B j ∈ S n , j =1... n} (600)such that, for some 0≤i≤n and scalars {t j , j =1... i} ,X ⋆ +i∑t j B j (601)j=1becomes an extreme point of A ∩ S n + and remains an optimal solution of(546P). Membership of (601) to affine subset A is secured for the i thperturbation by demanding〈B i , A j 〉 = 0, j =1... m (602)while membership to the positive semidefinite cone S n + is insured by smallperturbation (611). In this manner feasibility is insured. Optimality is provedin4.3.3.The following simple algorithm has very low computational intensity andlocates an optimal extreme point, assuming a nontrivial solution:4.3.1.0.1 Procedure. Rank reduction. (F.4)initialize: B i = 0 ∀ifor iteration i=1...n{1. compute a nonzero perturbation matrix B i of X ⋆ + i−1 ∑j=1t j B j2. maximize t isubject to X ⋆ + i ∑j=1t j B j ∈ S n +}