v2009.01.01 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 513
This solution is closed form.
This solution is equivalent to projection on a polyhedral cone in
the spectral domain (spectral projection, projection on a spectral
cone7.1.3.0.1); a necessary and sufficient condition (A.3.1) for
membership of a symmetric matrix to a rank ρ subset of a positive
semidefinite cone (2.9.2.1).
Because U ⋆ = Q , a minimum-distance projection on a rank ρ subset
of the positive semidefinite cone is a positive semidefinite matrix
orthogonal (in the Euclidean sense) to direction of projection. 7.10
For the convex case problem, this solution is always unique. Otherwise,
distinct eigenvalues (multiplicity 1) in Λ guarantees uniqueness of this
solution by the reasoning inA.5.0.1 . 7.11
7.1.5 Problem 1 in spectral norm, convex case
When instead we pose the matrix 2-norm (spectral norm) in Problem 1 (1219)
for the convex case ρ = N −1, then the new problem
minimize ‖−VN T(D − H)V N ‖ 2
D
(1249)
subject to D ∈ EDM N
is convex although its solution is not necessarily unique; 7.12 giving rise to
nonorthogonal projection (E.1) on the positive semidefinite cone S N−1
+ .
Indeed, its solution set includes the Frobenius solution (1221) for the convex
case whenever −VN THV N is a normal matrix. [158,1] [151] [53,8.1.1]
Singular value problem (1249) is equivalent to
minimize µ
µ , D
subject to −µI ≼ −VN T(D − H)V N ≼ µI (1250)
D ∈ EDM N
7.10 But Theorem E.9.2.0.1 for unique projection on a closed convex cone does not apply
here because the direction of projection is not necessarily a member of the dual PSD cone.
This occurs, for example, whenever positive eigenvalues are truncated.
7.11 Uncertainty of uniqueness prevents the erroneous conclusion that a rank ρ subset
(198) were a convex body by the Bunt-Motzkin
[
theorem
]
(E.9.0.0.1).
2 0
7.12 For each and every |t|≤2, for example, has the same spectral-norm value.
0 t