v2007.09.17 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 257Optimal matrix W ⋆ is defined as any direction matrix yielding optimalsolution G ⋆ of rank n or less to then convex equivalent (632) of feasibilityproblem (631); id est, any direction matrix for which the last N − neigenvalues λ of G ⋆ are zero: (p.541)N∑λ(G ⋆ ) i = 〈G ⋆ , W ⋆ 〉 = ∆ 0 (633)i=n+1We emphasize that convex problem (632) is not a relaxation of therank-constrained feasibility problem (631); at convergence, convex iteration(632) (1480a) makes it instead an equivalent problem. 4.21We make no assumption regarding uniqueness of direction matrix W .The feasible set of direction matrices in (1480a) is the convex hull of outerproduct of all rank-(N − n) orthonormal matrices; videlicet,conv { UU T | U ∈ R N×N−n , U T U = I } = { A∈ S N | I ≽ A ≽ 0, 〈I , A 〉= N − n } (79)Set {UU T | U ∈ R N×N−n , U T U = I} comprises the extreme points of thisFantope (79).4.4.1.1 convergenceWe study convergence to ascertain conditions under which a direction matrixwill reveal a feasible G matrix of rank n or less in semidefinite program (632).Denote by W ⋆ a particular optimal direction matrix from semidefiniteprogram (1480a) such that (633) holds. Then we define global convergenceof the iteration (632) (1480a) to correspond with this vanishing vectorinner-product (633) of optimal solutions.Because this iterative technique for constraining rank is not a projectionmethod, it can find a rank-n solution G ⋆ ((633) will be satisfied) only if atleast one exists in the feasible set of program (632).4.4.1.1.1 Proof. Suppose 〈G ⋆ , W 〉=φ is satisfied for somenonnegative constant φ after any particular iteration (632) (1480a) of thetwo minimization problems. Once a particular value of φ is achieved,4.21 Terminology equivalent problem meaning, optimal solution to one problem can bederived from optimal solution to another. Terminology same problem means: optimalsolution set for one problem is identical to the optimal solution set of another (withouttransformation).