v2009.01.01 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 503
finds D to attain Euclidean distance of vectorized −VHV to the positive
semidefinite cone in ambient isometrically isomorphic R N(N+1)/2 , whereas
minimize ‖−VN T(D − H)V N ‖ 2 F
D
(1217)
subject to D ∈ EDM N
attains Euclidean distance of vectorized −VN THV N to the positive
semidefinite cone in isometrically isomorphic subspace R N(N−1)/2 ; quite
different projections 7.4 regardless of whether affine dimension is constrained.
But substitution of auxiliary matrix VW T (B.4.3) or V † N
yields the same
result as (1215.1) 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
(1218)
We see no compelling reason to prefer one particular auxiliary V -matrix
over another. Each has its own coherent interpretations; e.g.,5.4.2,6.7.
Neither can we say any particular problem formulation produces generally
better results than another.
7.1 First prevalent problem:
Projection on PSD cone
This first problem
minimize ‖−VN T(D − H)V ⎫
N ‖ 2 F ⎪⎬
D
subject to rankVN TDV N ≤ ρ Problem 1 (1219)
⎪
D ∈ EDM N ⎭
7.4 The isomorphism T(Y )=V †T
N Y V † N onto SN c = {V X V | X ∈ S N } relates the map in
(1217) to that in (1216), but is not an isometry.