v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
280 CHAPTER 4. SEMIDEFINITE PROGRAMMINGC0PΓ 1Γ 2S+3A=∂HFigure 81: Visualizing positive semidefinite cone in high dimension: Properpolyhedral cone S 3 + ⊂ R 3 representing positive semidefinite cone S 3 + ⊂ S 3 ;analogizing its intersection with hyperplane S 3 + ∩ ∂H . Number of facets isarbitrary (analogy is not inspired by eigenvalue decomposition). The rank-0positive semidefinite matrix corresponds to the origin in R 3 , rank-1 positivesemidefinite matrices correspond to the edges of the polyhedral cone, rank-2to the facet relative interiors, and rank-3 to the polyhedral cone interior.Vertices Γ 1 and Γ 2 are extreme points of polyhedron P =∂H ∩ S 3 + , andextreme directions of S 3 + . A given vector C is normal to another hyperplane(not illustrated but independent w.r.t ∂H) containing line segment Γ 1 Γ 2minimizing real linear function 〈C, X〉 on P . (confer Figure 26, Figure 30)
4.1. CONIC PROBLEM 281faces of S 3 + correspond to faces of S 3 + (confer Table 2.9.2.3.1):k dim F(S+) 3 dim F(S 3 +) dim F(S 3 + ∋ rank-k matrix)0 0 0 0boundary 1 1 1 12 2 3 3interior 3 3 6 6Integer k indexes k-dimensional faces F of S 3 + . Positive semidefinitecone S 3 + has four kinds of faces, including cone itself (k = 3,boundary + interior), whose dimensions in isometrically isomorphicR 6 are listed under dim F(S 3 +). Smallest face F(S 3 + ∋ rank-k matrix)that contains a rank-k positive semidefinite matrix has dimensionk(k + 1)/2 by (222).For A equal to intersection of m hyperplanes having independentnormals, and for X ∈ S 3 + ∩ A , we have rankX ≤ m ; the analogueto (272).Proof. With reference to Figure 81: Assume one (m = 1) hyperplaneA = ∂H intersects the polyhedral cone. Every intersecting planecontains at least one matrix having rank less than or equal to 1 ; id est,from all X ∈ ∂H ∩ S 3 + there exists an X such that rankX ≤ 1. Rank 1is therefore an upper bound in this case.Now visualize intersection of the polyhedral cone with two (m = 2)hyperplanes having linearly independent normals. The hyperplaneintersection A makes a line. Every intersecting line contains at least onematrix having rank less than or equal to 2, providing an upper bound.In other words, there exists a positive semidefinite matrix X belongingto any line intersecting the polyhedral cone such that rankX ≤ 2.In the case of three independent intersecting hyperplanes (m = 3), thehyperplane intersection A makes a point that can reside anywhere inthe polyhedral cone. The upper bound on a point in S+ 3 is also thegreatest upper bound: rankX ≤ 3.
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280 CHAPTER 4. SEMIDEFINITE PROGRAMMINGC0PΓ 1Γ 2S+3A=∂HFigure 81: Visualizing positive semidefinite cone in high dimension: Properpolyhedral cone S 3 + ⊂ R 3 representing positive semidefinite cone S 3 + ⊂ S 3 ;analogizing its intersection with hyperplane S 3 + ∩ ∂H . Number of facets isarbitrary (analogy is not inspired by eigenvalue decomposition). The rank-0positive semidefinite matrix corresponds to the origin in R 3 , rank-1 positivesemidefinite matrices correspond to the edges of the polyhedral cone, rank-2to the facet relative interiors, and rank-3 to the polyhedral cone interior.Vertices Γ 1 and Γ 2 are extreme points of polyhedron P =∂H ∩ S 3 + , andextreme directions of S 3 + . A given vector C is normal to another hyperplane(not illustrated but independent w.r.t ∂H) containing line segment Γ 1 Γ 2minimizing real linear function 〈C, X〉 on P . (confer Figure 26, Figure 30)