v2007.09.13 - Convex Optimization

7.1. FIRST PREVALENT PROBLEM: 4517.1.3.0.4 Corollary. Monotone nonnegative sort.Any given vectors σ,γ∈R N−1 satisfy a tight Euclidean distance inequality‖π(σ) − π(γ)‖ ≤ ‖σ − γ‖ (1132)where nonlinear function π(γ) sorts vector γ into nonincreasing orderthereby providing the least lower bound with respect to every possiblesorting.⋄Given γ ∈ R N−1infσ∈R N−1+‖σ−γ‖ = infσ∈R N−1+‖π(σ)−π(γ)‖ = infσ∈R N−1+‖σ−π(γ)‖ = inf ‖σ−π(γ)‖σ∈K M+Yet for γ representing an arbitrary vector of eigenvalues, because(1133)inf − γ‖0‖σ 2 ≥ inf − π(γ)‖σ∈R ρ +0‖σ 2 =σ∈R ρ +infσ∈K ρ M+0‖σ − π(γ)‖ 2 (1134)then projection of γ on the eigenspectra of a rank ρ subset can be tightenedsimply by presorting γ into nonincreasing order.Proof. Simply because π(γ) 1:ρ ≽ π(γ 1:ρ )inf − γ‖0‖σ 2 = γρ+1:N−1 T γ ρ+1:N−1 + inf ‖σ 1:ρ − γ 1:ρ ‖ 2σ∈R N−1+= γ T γ + inf σ1:ρσ T 1:ρ − 2σ1:ργ T 1:ρσ∈R ρ +inf − γ‖0‖σ 2 ≥ infσ∈R ρ +σ∈R N−1+σ1:ρσ T 1:ρ − 2σ T (1135)1:ρπ(γ) 1:ρσ∈R N−1+− π(γ)‖σ∈R ρ +0‖σ 2≥ γ T γ + inf