v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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560 CHAPTER 7. PROXIMITY PROBLEMSof an ordered vector of eigenvalues (in diagonal matrix Λ) on a subset of themonotone nonnegative cone (2.13.9.4.2)K M+ = {υ | υ 1 ≥ υ 2 ≥ · · · ≥ υ N−1 ≥ 0} ⊆ R N−1+ (429)Of interest, momentarily, is only the smallest convex subset of themonotone nonnegative cone K M+ containing every nonincreasingly orderedeigenspectrum corresponding to a rank ρ subset of the positive semidefinitecone S N−1+ ; id est,K ρ M+ {υ ∈ Rρ | υ 1 ≥ υ 2 ≥ · · · ≥ υ ρ ≥ 0} ⊆ R ρ + (1326)a pointed polyhedral cone, a ρ-dimensional convex subset of themonotone nonnegative cone K M+ ⊆ R N−1+ having property, for λ denotingeigenspectra,[ Kρ]M+= π(λ(rank ρ subset)) ⊆ K N−10M+ K M+ (1327)For each and every elemental eigenspectrumγ ∈ λ(rank ρ subset)⊆ R N−1+ (1328)of the rank ρ subset (ordered or unordered in λ), there is a nonlinearsurjection 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+ containingevery ordered eigenspectrum corresponding to the rank ρ subset of a positivesemidefinite cone (2.9.2.1).7.1.3.0.3 Proposition. (Hardy-Littlewood-Pólya) Inequalities. [179,X][55,1.2] Any vectors σ and γ in R N−1 satisfy a tight inequalityπ(σ) T π(γ) ≥ σ T γ ≥ π(σ) T Ξπ(γ) (1329)where Ξ is the order-reversing permutation matrix defined in (1728),and permutator π(γ) is a nonlinear function that sorts vector γ intononincreasing order thereby providing the greatest upper bound and leastlower bound with respect to every possible sorting.⋄

7.1. FIRST PREVALENT PROBLEM: 5617.1.3.0.4 Corollary. Monotone nonnegative sort.Any given vectors σ,γ∈R N−1 satisfy a tight Euclidean distance inequality‖π(σ) − π(γ)‖ ≤ ‖σ − γ‖ (1330)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(1331)inf[Rρσ∈ +0]‖σ − γ‖ 2 ≥ inf[Rρσ∈ +0]‖σ − π(γ)‖ 2 =inf[ Kρσ∈ M+0]‖σ − π(γ)‖ 2 (1332)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[Rρσ∈ +0inf[Rρσ∈ +0]‖σ − γ‖ 2 = γ T ρ+1:N−1 γ ρ+1:N−1 + inf]‖σ − γ‖ 2 ≥ inf= γ T γ + infσ∈R N−1+≥ γ T γ + infσ∈R N−1+σ T 1:ρσ 1:ρ − 2σ T 1:ργ 1:ρ‖σ 1:ρ − γ 1:ρ ‖ 2σ1:ρσ T 1:ρ − 2σ T (1333)1:ρπ(γ) 1:ρσ∈R N−1+[Rρ]‖σ − π(γ)‖ 2σ∈ +0

7.1. FIRST PREVALENT PROBLEM: 5617.1.3.0.4 Corollary. Monotone nonnegative sort.Any given vectors σ,γ∈R N−1 satisfy a tight Euclidean distance inequality‖π(σ) − π(γ)‖ ≤ ‖σ − γ‖ (1330)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(1331)inf[Rρσ∈ +0]‖σ − γ‖ 2 ≥ inf[Rρσ∈ +0]‖σ − π(γ)‖ 2 =inf[ Kρσ∈ M+0]‖σ − π(γ)‖ 2 (1332)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[Rρσ∈ +0inf[Rρσ∈ +0]‖σ − γ‖ 2 = γ T ρ+1:N−1 γ ρ+1:N−1 + inf]‖σ − γ‖ 2 ≥ inf= γ T γ + infσ∈R N−1+≥ γ T γ + infσ∈R N−1+σ T 1:ρσ 1:ρ − 2σ T 1:ργ 1:ρ‖σ 1:ρ − γ 1:ρ ‖ 2σ1:ρσ T 1:ρ − 2σ T (1333)1:ρπ(γ) 1:ρσ∈R N−1+[Rρ]‖σ − π(γ)‖ 2σ∈ +0

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