v2006.03.09 - Convex Optimization

346 CHAPTER 5. EDM CONEcone; id est, the optimal solution of 5.8minimize ‖D ◦ − H‖ FD ◦ ∈ S Nsubject to D ◦ − δ(D ◦ 1) ≽ 0(825)has dual affine dimension complementary to affine dimension correspondingto the optimal solution ofminimize ‖D − H‖ FD∈S N hsubject to −VN TDV N ≽ 0(826)Precisely,rank(D ◦⋆ −δ(D ◦⋆ 1)) + rank(V T ND ⋆ V N ) = N −1 (827)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 (828)When low affine dimension is a desirable result of projection on theEDM cone, projection on the polar EDM cone should be performed instead.Convex polar problem (825) can be solved for D ◦⋆ by transforming to anequivalent Schur-form semidefinite program (A.4.1). Interior-point methodsfor numerically solving semidefinite programs tend to produce high-ranksolutions. (6.1.1) Then D ⋆ = H − D ◦⋆ ∈ EDM N by Corollary E.9.2.2.1, andD ⋆ will tend to have low affine dimension. This approach breaks whenattempting projection on a cone subset discriminated by affine dimensionor rank, because then we have no complementarity relation like (827) or(828) (7.1.4.1).5.8 This dual projection can be solved quickly (without semidefinite programming) viaLemma 5.8.1.1.1; 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 N c(H − δ(D ◦⋆ 1))Foreknowledge of an optimal solution D ◦⋆ as argument to projection suggests recursion.