v2010.10.26 - Convex Optimization

4.8. CONVEX ITERATION RANK-1 393the latter providing direction of search W for a rank-1 matrix G in (876).An optimal solution to (877) has closed form (p.658). These convex problems(876) (877) are iterated with (872) until a rank-1 G matrix is found and theobjective of (876) vanishes.For a fixed number of equality constraints m and upper bound ρ , thefeasible set in rank-constrained semidefinite feasibility problem (869) growswith matrix dimension n . Empirically, this semidefinite feasibility problembecomes easier to solve by method of convex iteration as n increases. Wefind a good value of weight w ≈ 5 for randomized problems. Initial value ofdirection matrix W is not critical. With dimension n = 26 and numberof equality constraints m = 10, convergence to a rank-ρ = 2 solution tosemidefinite feasibility problem (869) is achieved for nearly all realizationsin less than 10 iterations. 4.64 For small n , stall detection is required; onenumerical implementation is disclosed on Wıκımization.This diagonalization decomposition technique is extensible to otherproblem types, of course, including optimization problems having nontrivialobjectives. Because of a greatly expanded feasible set, for example,find X ∈ S nsubject to A svec X ≽ bX ≽ 0rankX ≤ ρ(878)this relaxation of (869) [225,III] is more easily solved by convex iterationrank-1 . 4.65 Because of the nonconvex nature of a rank-constrained problem,more generally, there can be no proof of global convergence of convex iterationfrom an arbitrary initialization; although, local convergence is guaranteed byvirtue of monotonically nonincreasing real objective sequences. [258,1.2][42,1.1]4.64 This finding is significant in so far as Barvinok’s Proposition 2.9.3.0.1 predictsexistence of matrices having rank 4 or less in this intersection, but his upper bound canbe no tighter than 3.4.65 The inequality in A remains in the constraints after diagonalization.