v2007.09.13 - Convex Optimization

450 CHAPTER 7. PROXIMITY PROBLEMSof an ordered vector of eigenvalues (in diagonal matrix Λ) on a subset of themonotone nonnegative cone (2.13.9.4.1)K M+ = {υ | υ 1 ≥ υ 2 ≥ · · · ≥ υ N−1 ≥ 0} ⊆ R N−1+ (370)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 ∈ R ρ | υ 1 ≥ υ 2 ≥ · · · ≥ υ ρ ≥ 0} ⊆ R ρ + (1128)a pointed polyhedral cone, a ρ-dimensional convex subset of themonotone nonnegative cone K M+ ⊆ R N−1+ having property, for λ denotingeigenspectra,[ρ] KM+= π(λ(rank ρ subset)) ⊆ K N−1 ∆0M+= K M+ (1129)For each and every elemental eigenspectrumγ ∈ λ(rank ρ subset)⊆ R N−1+ (1130)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) [128,X][41,1.2] 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) (1131)where Ξ is the order-reversing permutation matrix defined in (1502),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.⋄