v2009.01.01 - Convex Optimization

508 CHAPTER 7. PROXIMITY PROBLEMS
projection of an ordered vector of eigenvalues (in diagonal matrix Λ) on a
subset of the monotone nonnegative cone (2.13.9.4.2)
K M+ = {υ | υ 1 ≥ υ 2 ≥ · · · ≥ υ N−1 ≥ 0} ⊆ R N−1
+ (383)
Of interest, momentarily, is only the smallest convex subset of the
monotone nonnegative cone K M+ containing every nonincreasingly ordered
eigenspectrum corresponding to a rank ρ subset of the positive semidefinite
cone S N−1
+ ; id est,
K ρ M+
∆
= {υ ∈ R ρ | υ 1 ≥ υ 2 ≥ · · · ≥ υ ρ ≥ 0} ⊆ R ρ + (1231)
a pointed polyhedral cone, a ρ-dimensional convex subset of the
monotone nonnegative cone K M+ ⊆ R N−1
+ having property, for λ denoting
eigenspectra,
[
ρ
] K
M+
= π(λ(rank ρ subset)) ⊆ K N−1 ∆
0
M+
= K M+ (1232)
For each and every elemental eigenspectrum
γ ∈ λ(rank ρ subset)⊆ R N−1
+ (1233)
of the rank ρ subset (ordered or unordered in λ), there is a nonlinear
surjection π(γ) onto K ρ M+ .
7.1.3.0.2 Exercise. Smallest spectral cone.
Prove that there is no convex subset of K M+ smaller than K ρ M+ containing
every ordered eigenspectrum corresponding to the rank ρ subset of a positive
semidefinite cone (2.9.2.1).
7.1.3.0.3 Proposition. (Hardy-Littlewood-Pólya) Inequalities. [153,X]
[48,1.2] Any vectors σ and γ in R N−1 satisfy a tight inequality
π(σ) T π(γ) ≥ σ T γ ≥ π(σ) T Ξπ(γ) (1234)
where Ξ is the order-reversing permutation matrix defined in (1608),
and permutator π(γ) is a nonlinear function that sorts vector γ into
nonincreasing order thereby providing the greatest upper bound and least
lower bound with respect to every possible sorting.
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