v2010.10.26 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 309whose feasible set is a Fantope (2.3.2.0.1), and where G ⋆ is an optimalsolution to problem (736) given some iterate W . The idea is to iteratesolution of (736) and (1700a) until convergence as defined in4.4.1.2: 4.24N∑λ(G ⋆ ) i = 〈G ⋆ , W ⋆ 〉 = λ(G ⋆ ) T λ(W ⋆ ) 0 (737)i=n+1defines global convergence of the iteration; a vanishing objective that is acertificate of global optimality but cannot be guaranteed. Optimal directionvector W ⋆ is defined as any positive semidefinite matrix yielding optimalsolution G ⋆ of rank n or less to then convex equivalent (736) of feasibilityproblem (735):(735)findG∈S NGsubject to G ∈ CG ≽ 0rankG ≤ n≡minimizeG∈S N 〈G , W ⋆ 〉subject to G ∈ CG ≽ 0id est, any direction vector for which the last N − n nonincreasingly orderedeigenvalues λ of G ⋆ are zero. In any semidefinite feasibility problem, asolution of lowest rank must be an extreme point of the feasible set. Thenthere must exist a linear objective function such that this lowest-rank feasiblesolution optimizes the resultant SDP. [239,2.4] [240]We emphasize that convex problem (736) is not a relaxation ofrank-constrained feasibility problem (735); at global convergence, convexiteration (736) (1700a) makes it instead an equivalent problem.4.4.1.1 direction matrix interpretation(confer4.5.1.2) The feasible set of direction matrices in (1700a) is the convexhull of outer product of all rank-(N − n) orthonormal matrices; videlicet,(736)conv { UU T | U ∈ R N×N−n , U T U = I } = { A∈ S N | I ≽ A ≽ 0, 〈I , A 〉= N − n } (90)This set (92), argument to conv{ } , comprises the extreme points of thisFantope (90). An optimal solution W to (1700a), that is an extreme point,4.24 Proposed iteration is neither dual projection (Figure 164) or alternating projection(Figure 167). Sum of eigenvalues follows from results of Ky Fan (page 658). Innerproduct of eigenvalues follows from (1590) and properties of commutative matrix products(page 605).