v2009.01.01 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 123
2.9.2.7.1 Exercise. Projection on open convex cones.
Prove (236) using Theorem E.9.2.0.1.
Because each H ∈ S M has unique projection on S M +(ρ) (despite possibility
of repeated eigenvalues in Λ), we may conclude it is a convex set by the
Bunt-Motzkin theorem (E.9.0.0.1).
Compare (236) to the well-known result regarding Euclidean projection
on a rank ρ subset of the positive semidefinite cone (2.9.2.1)
S M + \ S M +(ρ + 1) = {X ∈ S M + | rankX ≤ ρ} (237)
P S M
+ \S M + (ρ+1) H = QΥ ⋆ Q T (238)
As proved in7.1.4, this projection of H corresponds to the eigenvalue map
Υ ⋆ ii =
{ max {0 , Λii } , i=1... ρ
0 , i=ρ+1... M
(1246)
Together these two results (236) and (1246) mean: A higher-rank solution
to projection on the positive semidefinite cone lies arbitrarily close to any
given lower-rank projection, but not vice versa. Were the number of
nonnegative eigenvalues in Λ known a priori not to exceed ρ , then these
two different projections would produce identical results in the limit ǫ→0.
2.9.2.8 Uniting constituents
Interior of the PSD cone int S M + is convex by Theorem 2.9.2.6.3, for example,
because all positive semidefinite matrices having rank M constitute the cone
interior.
All positive semidefinite matrices of rank less than M constitute the cone
boundary; an amalgam of positive semidefinite matrices of different rank.
Thus each nonconvex subset of positive semidefinite matrices, for 0