v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 201[30,1.1] Expression (479) for optimal projector W holds at each iteration,therefore ‖x ⋆ ‖ 2 (1 − (1 + ǫ) −1 ) must also represent the optimal objectivevalue f ⋆ at convergence.(477)Because the objective f (476) from problem (476) is also bounded belowby 0 on the same domain, this convergent optimal objective value f ⋆ (for (477)positive ǫ arbitrarily close to 0) is necessarily optimal for (476); id est,by (1457), andf ⋆ (477) ≥ f⋆ (476) ≥ 0 (481)lim = 0 (482)ǫ→0 +f⋆ (477)Since optimal (x ⋆ , U ⋆ ) from problem (477) is feasible to problem (476), andbecause their objectives are equivalent for projectors by (473), then converged(x ⋆ , U ⋆ ) must also be optimal to (476) in the limit. Because problem (476)is convex, this represents a globally optimal solution.3.1.7.2 Semidefinite program via SchurSchur complement (1307) can be used to convert a projection problemto an optimization problem in epigraph form. Suppose, for example,we are presented with the constrained projection problem studied byHayden & Wells in [132] (who provide analytical solution): Given A∈ R M×Mand some full-rank matrix S ∈ R M×L with L < Mminimize ‖A − X‖ 2X∈ S MFsubject to S T XS ≽ 0(483)Variable X is constrained to be positive semidefinite, but only on a subspacedetermined by S . First we write the epigraph form:minimize tX∈ S M , t∈Rsubject to ‖A − X‖ 2 F ≤ tS T XS ≽ 0(484)