v2006.03.09 - Convex Optimization

4.13. RECONSTRUCTION EXAMPLES 295New problem (696b) finds the unique minimum-distance projection of Ξdon the monotone nonnegative cone K M+ . By definingY †T ∆ = [e 1 − e 2 e 2 −e 3 e 3 −e 4 · · · e m ] ∈ R m×m (348)where m=N(N ∆ −1)/2, we may rewrite (696b) as an equivalent quadraticprogram; a convex optimization problem [38,4] in terms of thehalfspace-description of K M+ :minimize (σ − Ξd) T (σ − Ξd)σsubject to Y † σ ≽ 0(699)This quadratic program can be converted to a semidefinite program via Schurform (A.4.1); we get the equivalent problemminimizet∈R , σsubject tot[tI σ − Ξd(σ − Ξd) T 1]≽ 0(700)Y † σ ≽ 04.13.2.3 ConvergenceInE.10 we discuss convergence of alternating projection on intersectingconvex sets in a Euclidean vector space; convergence to a point in theirintersection. Here the situation is different for two reasons:Firstly, sets of positive semidefinite matrices having an upper bound onrank are generally not convex. Yet in7.1.4.0.1 we prove (696a) is equivalentto a projection of nonincreasingly ordered eigenvalues on a subset of thenonnegative orthant:minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3D ∈ EDM N≡minimize ‖Υ − Λ‖ FΥ [ ]R3+ (701)subject to δ(Υ) ∈0∆where −VN TDV N =UΥU T ∈ S N−1 and −VN TOV N =QΛQ T ∈ S N−1 areordered diagonalizations (A.5). It so happens: optimal orthogonal U ⋆always equals Q given. Linear operator T(A) = U ⋆T AU ⋆ , acting on square∆