v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 363where x ⋆ i ∈ {−1, 1}, i=1... N . The two variables X and x are madedependent via their assignment to rank-1 matrix G . By (1620), an optimalrank-1 matrix G ⋆ must take the form (809).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(811)Solution of this semidefinite program is iterated with calculation of thedirection matrix Y from semidefinite program (783). At convergence, in thesense (737), convex problem (811) becomes equivalent to nonconvex Booleanproblem (808).Direction matrix Y can be an orthogonal projector having closed-formexpression, by (1700a), although convex iteration is not a projection method.(4.4.1.1) Given randomized data A and b for a large problem, we find thatstalling becomes likely (convergence of the iteration to a positive objective〈G ⋆ , Y 〉). To overcome this behavior, we introduce a heuristic into theimplementation on Wıκımization that momentarily reverses direction ofsearch (like (784)) upon stall detection. We find that rate of convergencecan be sped significantly by detecting stalls early.4.6.0.0.9 Example. Variable-vector normalization.Suppose, within some convex optimization problem, we want vector variablesx, y ∈ R N constrained by a nonconvex equality:x‖y‖ = y (812)id est, ‖x‖ = 1 and x points in the same direction as y≠0 ; e.g.,minimize f(x, y)x , ysubject to (x, y) ∈ Cx‖y‖ = y(813)where f is some convex function and C is some convex set. We can realizethe nonconvex equality by constraining rank and adding a regularization