v2007.11.26 - Convex Optimization

256 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 (550). Then we proceed withrank reduction of X ⋆ as though the semidefinite program were in prototypicalform (547P).4.4 Rank-constrained semidefinite programHere we introduce a technique for finding low-rank optimal solutions tosemidefinite programs 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(632)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 is equivalentlyexpressed with convex constraints:minimize 〈G , W 〉G∈S N , W ∈ S Nsubject to G ∈ CG ≽ 00 ≼ W ≼ ItrW = N − n(633)This we call the underlying problem which is very difficult to solve becauseof the bilinear objective function. So instead we solve the convex problemminimizeG∈S N 〈G , W 〉subject to G ∈ CG ≽ 0(634)