v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 125submatrix ΦAΦ∈ S M + rank ΦAΦ ≤ rank Φ , apply ordered diagonalizationinstead toˆΦ T A ˆΦ = UΥU T ∈ S rankΦ+ (226)ThenF ( S M + ∋ΦAΦ ) = {X ∈ S M + | N(X) ⊇ N(ΦAΦ)}= {X ∈ S M + | 〈ˆΦU(I − ΥΥ † )U TˆΦT + I − Φ , X〉 = 0}= {ˆΦUΥΥ † ΨΥΥ † U TˆΦT | Ψ ∈ S rankΦ+ }≃ S rankΦAΦ+(227)where binary diagonal matrix Φ is partitioned into nonzero and zero columnsby permutation Ξ∈ R M×M ;ΦΞ T [ ˆΦ 0 ]∈ R M×M , rank ˆΦ = rank Φ , Φ = ˆΦˆΦ T ∈ S M , ˆΦTˆΦ = I (228)Any embedded principal submatrix may be expressedΦAΦ = ˆΦˆΦ T A ˆΦˆΦ T ∈ S M + (229)where ˆΦ T A ˆΦ∈ S rankΦ+ extracts the principal submatrix whereas ˆΦˆΦ T A ˆΦˆΦ Tembeds it.2.9.2.5.1 Example. Smallest face containing disparate elements.Imagine two vectorized matrices A 1 and A 2 on diametrically opposed sidesof the positive semidefinite cone boundary pictured in Figure 43. Smallestface formula (221) can be altered to accommodate a union of points:F(S M + ⊃ ⋃ iA i)={X ∈ S M +∣ N(X) ⊇ ⋂ iN(A i )}(230)To see that, regard svecA 1 as normal to a hyperplane in R 3 containing avectorized basis for its nullspace: svec basis N(A 1 ) (2.5.3). Similarly, thereis a second hyperplane containing svec basis N(A 2 ) having normal svecA 2 .While each hyperplane is two-dimensional, each nullspace has only oneaffine dimension because A 1 and A 2 are rank-1. Because our interest isonly that part of the nullspace in the positive semidefinite cone, then forX,A i ∈ S M + by〈X , A i 〉 = 0 ⇔ XA i = A i X = 0 (1590)