v2007.09.13 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 279Then design an equivalent semidefinite feasibility problem to find a Booleansolution to Ax ≼ b :findX∈S Nx ∈ R Nsubject to Ax ≼ bG =[ X xx T 1rankG = 1(G ≽ 0)δ(X) = 1](667)where x ⋆ i ∈ {−1, 1}, i=1... N . The two variables X and x are madedependent via their assignment to rank-1 matrix G . By (1399), an optimalrank-1 matrix G ⋆ must take the form (666).As before, we regularize the rank constraint by introducing a directionmatrix Y into the objective:minimize 〈G, Y 〉X∈S N , x∈R Nsubject to Ax ≼ b[ X xG =x T 1δ(X) = 1]≽ 0(668)Solution of this semidefinite program is iterated with calculation of thedirection matrix Y from semidefinite program (652). At convergence, in thesense (633), convex problem (668) becomes equivalent to nonconvex Booleanproblem (665).By (1475a), direction matrix Y can be an orthogonal projector havingclosed-form expression. Given randomized data A and b for a large problem,we find that stalling becomes likely (convergence of the iteration to a positivefixed point 〈G ⋆ , Y 〉). To overcome this behavior, we introduce a heuristicinto the implementation inF.6 that momentarily reverses direction of search(≈ −Y ) upon stall detection. We find that rate of convergence can be spedsignificantly by detecting stalls early.