v2007.09.13 - 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 (549). Then we proceed withrank reduction of X ⋆ as though the semidefinite program were in prototypicalform (546P).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 iterationGiven a feasibility problem of the formfind G ∈ S N +subject to G ∈ CrankG ≤ n(631)where C is a convex set presumed to contain positive semidefinite matricesof rank n or less, we instead solve the convex problemminimizeG∈S N 〈G , W 〉subject to G ∈ CG ≽ 0(632)where direction matrix W is an optimal solution to semidefinite programN∑λ(G ⋆ ) ii=n+1= minimizeW ∈ S N 〈G ⋆ , W 〉subject to 0 ≼ W ≼ ItrW = N − n(1475a)whose feasible set is a Fantope (2.3.2.0.1), and where G ⋆ is an optimalsolution to problem (632) given some iterate W . The idea is to iteratesolution of (632) and (1475a) until convergence, as defined in4.4.1.1. 4.204.20 The proposed iteration is not an alternating projection. (confer Figure 119)