v2010.10.26 - Convex Optimization

308 CHAPTER 4. SEMIDEFINITE PROGRAMMINGequality constraints. In other words, we take the union of active inequalityconstraints (as equalities) with equality constraints A svec X = b to forma composite affine subset  substituting for (652). Then we proceed withrank reduction of X ⋆ as though the semidefinite program were in prototypicalform (649P).4.4 Rank-constrained semidefinite programWe generalize the trace heuristic (7.2.2.1), for finding low-rank optimalsolutions to SDPs of a more general form:4.4.1 rank constraint by convex iterationConsider a semidefinite feasibility problem of the formfindG∈S NGsubject to G ∈ CG ≽ 0rankG ≤ n(735)where C is a convex set presumed to contain positive semidefinite matricesof rank n or less; id est, C intersects the positive semidefinite cone boundary.We propose that this rank-constrained feasibility problem can be equivalentlyexpressed as the convex problem sequence (736) (1700a):minimizeG∈S N 〈G , W 〉subject to G ∈ CG ≽ 0(736)where direction vector 4.23 W is an optimal solution to semidefinite program,for 0≤n≤N −1N∑λ(G ⋆ ) i = minimize 〈G ⋆ , W 〉 (1700a)i=n+1W ∈ S Nsubject to 0 ≼ W ≼ ItrW = N − n4.23 Search direction W is a hyperplane-normal pointing opposite to direction of movementdescribing minimization of a real linear function 〈G, W 〉 (p.75).