v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
106 CHAPTER 2. CONVEX GEOMETRYwhich is isomorphic with the convex cone S rank A+ . Thus dimension of thesmallest face containing given matrix A isdim F ( S M + ∋A ) = rank(A)(rank(A) + 1)/2 (191)in isomorphic R M(M+1)/2 , and each and every face of S M + is isomorphic witha positive semidefinite cone having dimension the same as the face. Observe:not all dimensions are represented, and the only zero-dimensional face is theorigin. The positive semidefinite cone has no facets, for example.2.9.2.3.1 Table: Rank k versus dimension of S 3 + facesk dim F(S 3 + ∋ rank-k matrix)0 0boundary 1 12 3interior 3 6For the positive semidefinite cone S 2 + in isometrically isomorphic R 3depicted in Figure 31, for example, rank-2 matrices belong to the interior ofthe face having dimension 3 (the entire closed cone), while rank-1 matricesbelong to the relative interior of a face having dimension 1 (the boundaryconstitutes all the one-dimensional faces, in this dimension, which are raysemanating from the origin), and the only rank-0 matrix is the point at theorigin (the zero-dimensional face).Any simultaneously diagonalizable positive semidefinite rank-k matricesbelong to the same face (190). That observation leads to the followinghyperplane characterization of PSD cone faces: Any rank-k
2.9. POSITIVE SEMIDEFINITE (PSD) CONE 107makes projectors Q(:,k+1:M)Q(:,k+1:M) T indexed by k , in other words,each projector describing a normal svec ( Q(:,k+1:M)Q(:,k+1:M) T) to asupporting hyperplane ∂H + (containing the origin) exposing (2.11) a faceof the positive semidefinite cone containing only rank-k matrices.2.9.2.4 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 the direct correspondence of exposed faces to faces of S M + , it followsthere is no one-dimensional face of the positive semidefinite cone that is nota ray emanating from the origin.Symmetric dyads constitute the set of all extreme directions: For M >0{yy T ∈ S M | y ∈ R M } ⊂ ∂S M + (193)this superset (confer (156)) of extreme directions for the positive semidefinitecone is, generally, a subset of the boundary. For two-dimensional matrices,(Figure 31){yy T ∈ S 2 | y ∈ R 2 } = ∂S 2 + (194)while for one-dimensional matrices, in exception, (2.7){yy T ∈ S | y≠0} = int S + (195)Each and every extreme direction yy T makes the same angle with theidentity matrix in isomorphic R M(M+1)/2 , dependent only on dimension;videlicet, 2.32(yy T , I) = arccos( )〈yy T , I〉 1= arccos √‖yy T ‖ F ‖I‖ F M∀y ∈ R M (196)2.32 Analogy with respect to the EDM cone is considered by Hayden & Wells et alii[133, p.162] where it is found: angle is not constant. The extreme directions of the EDMcone can be found in6.5.3.1 while the cone axis is −E =11 T − I (898).
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106 CHAPTER 2. CONVEX GEOMETRYwhich is isomorphic with the convex cone S rank A+ . Thus dimension of thesmallest face containing given matrix A isdim F ( S M + ∋A ) = rank(A)(rank(A) + 1)/2 (191)in isomorphic R M(M+1)/2 , and each and every face of S M + is isomorphic witha positive semidefinite cone having dimension the same as the face. Observe:not all dimensions are represented, and the only zero-dimensional face is theorigin. The positive semidefinite cone has no facets, for example.2.9.2.3.1 Table: Rank k versus dimension of S 3 + facesk dim F(S 3 + ∋ rank-k matrix)0 0boundary 1 12 3interior 3 6For the positive semidefinite cone S 2 + in isometrically isomorphic R 3depicted in Figure 31, for example, rank-2 matrices belong to the interior ofthe face having dimension 3 (the entire closed cone), while rank-1 matricesbelong to the relative interior of a face having dimension 1 (the boundaryconstitutes all the one-dimensional faces, in this dimension, which are raysemanating from the origin), and the only rank-0 matrix is the point at theorigin (the zero-dimensional face).Any simultaneously diagonalizable positive semidefinite rank-k matricesbelong to the same face (190). That observation leads to the followinghyperplane characterization of PSD cone faces: Any rank-k