v2010.10.26 - Convex Optimization

6.8. DUAL EDM CONE 5396.8.1.5 Affine dimension complementarityFrom6.8.1.3 we have, for some A∈ S N−1+ (confer (1280))δ(D ∗ 1) − D ∗ = V N AV T N ∈ S N c ∩ S N + (1282)if and only if D ∗ belongs to the dual EDM cone. Call rank(V N AV T N ) dualaffine dimension. Empirically, we find a complementary relationship in affinedimension between the projection of some arbitrary symmetric matrix H onthe polar EDM cone, EDM N◦ = −EDM N∗ , and its projection on the EDMcone; id est, the optimal solution of 6.13minimize ‖D ◦ − H‖ FD ◦ ∈ S Nsubject to D ◦ − δ(D ◦ 1) ≽ 0(1283)has dual affine dimension complementary to affine dimension correspondingto the optimal solution ofminimize ‖D − H‖ FD∈S N hsubject to −VN TDV N ≽ 0(1284)Precisely,rank(D ◦⋆ −δ(D ◦⋆ 1)) + rank(V T ND ⋆ V N ) = N −1 (1285)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 theselfdual positive semidefinite cone and its polar:rankP −S N+H + rankP S N+H = N (1286)6.13 This polar projection can be solved quickly (without semidefinite programming) viaLemma 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.