v2009.01.01 - Convex Optimization

510 CHAPTER 7. PROXIMITY PROBLEMS
7.1.3.1 Orthant is best spectral cone for Problem 1
This means unique minimum-distance projection of γ on the nearest
spectral member of the rank ρ subset is tantamount to presorting γ into
nonincreasing order. Only then does unique spectral projection on a subset
K ρ M+
of the monotone nonnegative cone become equivalent to unique spectral
projection on a subset R ρ + of the nonnegative orthant (which is simpler);
in other words, unique minimum-distance projection of sorted γ on the
nonnegative orthant in a ρ-dimensional subspace of R N is indistinguishable
from its projection on the subset K ρ M+
of the monotone nonnegative cone in
that same subspace.
7.1.4 Closest-EDM Problem 1, “nonconvex” case
Trosset’s proof of solution (1221), for projection on a rank ρ subset of the
positive semidefinite cone S N−1
+ , was algebraic in nature. [306,2] Here we
derive that known result but instead using a more geometric argument via
spectral projection on a polyhedral cone (subsuming the proof in7.1.1).
In so doing, we demonstrate how nonconvex Problem 1 is transformed to a
convex optimization:
7.1.4.0.1 Proof. Solution (1221), nonconvex case.
As explained in7.1.2, we may instead work with the more facile generic
problem (1228). With diagonalization of unknown
B ∆ = UΥU T ∈ S N−1 (1239)
given desired affine dimension 0 ≤ ρ ≤ N −1 and diagonalizable
A ∆ = QΛQ T ∈ S N−1 (1240)
having eigenvalues in Λ arranged in nonincreasing order, by (41) the generic
problem is equivalent to
minimize ‖B − A‖ 2
B∈S N−1
F
subject to rankB ≤ ρ
B ≽ 0
≡
minimize ‖Υ − R T ΛR‖ 2 F
R , Υ
subject to rank Υ ≤ ρ (1241)
Υ ≽ 0
R −1 = R T