v2007.09.13 - Convex Optimization

6.8. DUAL EDM CONE 429has dual affine dimension complementary to affine dimension correspondingto the optimal solution ofminimize ‖D − H‖ FD∈S N hsubject to −VN TDV N ≽ 0(1086)Precisely,rank(D ◦⋆ −δ(D ◦⋆ 1)) + rank(V T ND ⋆ V N ) = N −1 (1087)and rank(D ◦⋆ −δ(D ◦⋆ 1))≤N−1 because vector 1 is always in the nullspaceof rank’s argument. This is similar to the known result for projection on theself-dual positive semidefinite cone and its polar:rankP −S N+H + rankP S N+H = N (1088)When low affine dimension is a desirable result of projection on theEDM cone, projection on the polar EDM cone should be performedinstead. Convex polar problem (1085) can be solved for D ◦⋆ bytransforming to an equivalent Schur-form semidefinite program (3.1.7.2).Interior-point methods for numerically solving semidefinite programs tendto produce high-rank solutions. (4.1.1) Then D ⋆ = H − D ◦⋆ ∈ EDM N byCorollary E.9.2.2.1, and D ⋆ will tend to have low affine dimension. Thisapproach breaks when attempting projection on a cone subset discriminatedby affine dimension or rank, because then we have no complementarityrelation like (1087) or (1088) (7.1.4.1).Lemma 6.8.1.1.1; rewriting,minimize ‖(D ◦ − δ(D ◦ 1)) − (H − δ(D ◦ 1))‖ FD ◦ ∈ S Nsubject to D ◦ − δ(D ◦ 1) ≽ 0which is the projection of affinely transformed optimal solution H − δ(D ◦⋆ 1) on S N c ∩ S N + ;D ◦⋆ − δ(D ◦⋆ 1) = P S N+P S Nc(H − δ(D ◦⋆ 1))Foreknowledge of an optimal solution D ◦⋆ as argument to projection suggests recursion.