v2007.09.13 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 445finds D to attain Euclidean distance of vectorized −VHV to the positivesemidefinite cone in ambient isometrically isomorphic R N(N+1)/2 , whereasminimize ‖−VN T(D − H)V N ‖ 2 FD(1114)subject to D ∈ EDM Nattains Euclidean distance of vectorized −VN THV N to the positivesemidefinite cone in isometrically isomorphic subspace R N(N−1)/2 ; quitedifferent projections 7.3 regardless of whether affine dimension is constrained.But substitution of auxiliary matrix VW T (B.4.3) or V † Nyields the sameresult as (1112.1) because V = V W VW T = V N V † N; id est,‖−V (D − H)V ‖ 2 F = ‖−V W V T W (D − H)V W V T W ‖2 F = ‖−V T W (D − H)V W‖ 2 F= ‖−V N V † N (D − H)V N V † N ‖2 F = ‖−V † N (D − H)V N ‖ 2 F(1115)We see no compelling reason to prefer one particular auxiliary V -matrixover another. Each has its own coherent interpretations; e.g.,5.4.2,6.7.Neither can we say any particular problem formulation produces generallybetter results than another.7.1 First prevalent problem:Projection on PSD coneThis first problemminimize ‖−VN T(D − H)V ⎫N ‖ 2 F ⎪⎬Dsubject to rankVN TDV N ≤ ρ Problem 1 (1116)⎪D ∈ EDM N ⎭7.3 The isomorphism T(Y )=V †TN Y V † N onto SN c = {V X V | X ∈ S N } relates the map in(1114) to that in (1113), but is not an isometry.