v2007.09.13 - Convex Optimization

230 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.1.1.2 Coexistence of low- and high-rank solutions; analogyThat low-rank and high-rank optimal solutions {X ⋆ } of (546P) coexist maybe grasped with the following analogy: We compare a proper polyhedral coneS 3 + in R 3 (illustrated in Figure 62) 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 matricesint S+ 3 has three dimensionsboundary ∂S 3 + contains rank-0, rank-1, and rank-2 matricesboundary ∂S+ 3 contains 0-, 1-, and 2-dimensional facesthe only rank-0 matrix resides in the vertex at the originRank-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 } (550)is not a connected set.In any SDP feasibility problem, an SDP feasible solution with the lowestrank must be an extreme point of the feasible set. Thus, there must existan SDP objective function such that this lowest-rank feasible solutionuniquely optimizes it. −Ye, 2006Rank of a sum of members F +G in Lemma 2.9.2.6.1 and location ofa difference F −G in2.9.2.9.1 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