v2010.10.26 - Convex Optimization

4.1. CONIC PROBLEM 279is the affine subset from primal problem (649P).4.1.2.2 Coexistence of low- and high-rank solutions; analogyThat low-rank and high-rank optimal solutions {X ⋆ } of (649P) coexist maybe grasped with the following analogy: We compare a proper polyhedral coneS 3 + in R 3 (illustrated in Figure 81) to the positive semidefinite cone S 3 + inisometrically isomorphic R 6 , difficult to visualize. The analogy is good:int S 3 + is constituted by rank-3 matrices.int S 3 + has three dimensions.boundary ∂S 3 + contains rank-0, rank-1, and rank-2 matrices.boundary ∂S 3 + contains 0-, 1-, and 2-dimensional faces.the only rank-0 matrix resides in the vertex at the origin.Rank-1 matrices are in one-to-one correspondence with extremedirections of S 3 + and S 3 + . The set of all rank-1 symmetric matrices inthis dimension{G ∈ S3+ | rankG=1 } (653)is not a connected set.Rank of a sum of members F +G in Lemma 2.9.2.9.1 and location ofa difference F −G in2.9.2.12.1 similarly hold for S 3 + and S 3 + .Euclidean distance from any particular rank-3 positive semidefinitematrix (in the cone interior) to the closest rank-2 positive semidefinitematrix (on the boundary) is generally less than the distance to theclosest rank-1 positive semidefinite matrix. (7.1.2)distance from any point in ∂S 3 + to int S 3 + is infinitesimal (2.1.7.1.1).distance from any point in ∂S 3 + to int S 3 + is infinitesimal.