v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 127Define a set of all PSD matrices having fixed principal submatrix ΦAΦ{ [S Ξ ˆΦT T A ˆΦ]}BB T Ξ ≽ 0C ∣ B ∈ RrankΦ×M−rankΦ , C ∈ S M−rankΦ+ (233)SoF ( S M + ⊇ S ) = { X ∈ S M + | N(X) ⊇ N(S) }= {X ∈ S M + | 〈ˆΦU(I − ΥΥ † )U TˆΦT , X〉 = 0}{ [ ] [ ]}UΥΥ= Ξ T †0 ΥΥ0 T Ψ† U T 0I 0 T ΞI ∣ Ψ ∈ SM +M−rank Φ+rankΦAΦ≃ S+(234)In the case that ΦAΦ is a leading principal submatrix, then Ξ=I .2.9.2.7 Extreme directions of positive semidefinite coneBecause the positive semidefinite cone is pointed (2.7.2.1.2), there is aone-to-one correspondence of one-dimensional faces with extreme directionsin any dimension M ; id est, because of the cone faces lemma (2.8.0.0.1)and direct correspondence of exposed faces to faces of S M + , it follows: thereis no one-dimensional face of the positive semidefinite cone that is not a rayemanating from the origin.Symmetric dyads constitute the set of all extreme directions: For M >1{yy T ∈ S M | y ∈ R M } ⊂ ∂S M + (235)this superset of extreme directions (infinite in number, confer (187)) for thepositive semidefinite cone is, generally, a subset of the boundary. By theextremes theorem (2.8.1.1.1), the convex hull of extreme rays and the originis the positive semidefinite cone: (2.8.1.2.1){ ∞}∑conv{yy T ∈ S M | y ∈ R M } = b i z i zi T | z i ∈ R M , b≽0 = S M + (236)i=1For two-dimensional matrices (M =2, Figure 43){yy T ∈ S 2 | y ∈ R 2 } = ∂S 2 + (237)