v2007.09.13 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 455This solution is closed-form and the only method known for enforcinga constraint on rank of an EDM in projection problems such as (1112).This solution is equivalent to projection on a polyhedral cone inthe spectral domain (spectral projection, projection on a spectralcone7.1.3.0.1); a necessary and sufficient condition (A.3.1) formembership of a symmetric matrix to a rank ρ subset of a positivesemidefinite cone (2.9.2.1).Because U ⋆ = Q , a minimum-distance projection on a rank ρ subsetof the positive semidefinite cone is a positive semidefinite matrixorthogonal (in the Euclidean sense) to direction of projection. 7.9For the convex case problem, this solution is always unique. Otherwise,distinct eigenvalues (multiplicity 1) in Λ guarantees uniqueness of thissolution by the reasoning inA.5.0.1 . 7.107.1.5 Problem 1 in spectral norm, convex caseWhen instead we pose the matrix 2-norm (spectral norm) in Problem 1 (1116)for the convex case ρ = N −1, then the new problemminimize ‖−VN T(D − H)V N ‖ 2D(1146)subject to D ∈ EDM Nis convex although its solution is not necessarily unique; 7.11 giving rise tononorthogonal projection (E.1) on the positive semidefinite cone S N−1+ .Indeed, its solution set includes the Frobenius solution (1118) for the convexcase whenever −VN THV N is a normal matrix. [132,1] [126] [46,8.1.1]Singular value problem (1146) is equivalent tominimize µµ , Dsubject to −µI ≼ −VN T(D − H)V N ≼ µI (1147)D ∈ EDM N7.9 But Theorem E.9.2.0.1 for unique projection on a closed convex cone does not applyhere because the direction of projection is not necessarily a member of the dual PSD cone.This occurs, for example, whenever positive eigenvalues are truncated.7.10 Uncertainty of uniqueness prevents the erroneous conclusion that a rank ρ subset(185) were a convex body by the Bunt-Motzkin[theorem](E.9.0.0.1).2 07.11 For each and every |t|≤2, for example, has the same spectral-norm value.0 t