v2009.01.01 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 509
7.1.3.0.4 Corollary. Monotone nonnegative sort.
Any given vectors σ,γ∈R N−1 satisfy a tight Euclidean distance inequality
‖π(σ) − π(γ)‖ ≤ ‖σ − γ‖ (1235)
where nonlinear function π(γ) sorts vector γ into nonincreasing order
thereby providing the least lower bound with respect to every possible
sorting.
⋄
Given γ ∈ R N−1
inf
σ∈R N−1
+
‖σ−γ‖ = inf
σ∈R N−1
+
‖π(σ)−π(γ)‖ = inf
σ∈R N−1
+
‖σ−π(γ)‖ = inf ‖σ−π(γ)‖
σ∈K M+
Yet for γ representing an arbitrary vector of eigenvalues, because
(1236)
inf
[
R
ρ
σ∈ +
0
]‖σ − γ‖ 2 ≥ inf
[
R
ρ
σ∈ +
0
]‖σ − π(γ)‖ 2 =
inf
[ ρ K
σ∈ M+
0
]‖σ − π(γ)‖ 2 (1237)
then projection of γ on the eigenspectra of a rank ρ subset can be tightened
simply by presorting γ into nonincreasing order.
Proof. Simply because π(γ) 1:ρ ≽ π(γ 1:ρ )
inf
[
R
ρ
σ∈ +
0
inf
[
R
ρ
σ∈ +
0
]‖σ − γ‖ 2 = γ T ρ+1:N−1 γ ρ+1:N−1 + inf
]‖σ − γ‖ 2 ≥ inf
= γ T γ + inf
σ∈R N−1
+
≥ γ T γ + inf
σ∈R N−1
+
σ T 1:ρσ 1:ρ − 2σ T 1:ργ 1:ρ
‖σ 1:ρ − γ 1:ρ ‖ 2
σ1:ρσ T 1:ρ − 2σ T (1238)
1:ρπ(γ) 1:ρ
σ∈R N−1
+
[
R
ρ
]‖σ − π(γ)‖ 2
σ∈ +
0