v2007.11.26 - Convex Optimization

its membership to the EDM cone. The faces of the EDM cone are described,but still open is the question whether all its faces are exposed as theyare for the positive semidefinite cone. The Schoenberg criterion (753),relating the EDM cone and a positive semidefinite cone, is revealed tobe a discretized membership relation (dual generalized inequalities, a newFarkas’-like lemma) between the EDM cone and its ordinary dual, EDM N∗ .A matrix criterion for membership to the dual EDM cone is derived that issimpler than the Schoenberg criterion:D ∗ ∈ EDM N∗ ⇔ δ(D ∗ 1) − D ∗ ≽ 0 (1106)We derive a new concise equality of the EDM cone to two subspaces and apositive semidefinite cone;EDM N = S N h ∩ ( )S N⊥c − S N + (1100)In chapter 7, Proximity problems, we explore methods of solutionto a few fundamental and prevalent Euclidean distance matrix proximityproblems; the problem of finding that distance matrix closest, in some sense,to a given matrix H :29minimize ‖−V (D − H)V ‖ 2 FDsubject to rankV DV ≤ ρD ∈ EDM Nminimize ‖D − H‖ 2 FDsubject to rankV DV ≤ ρD ∈ EDM Nminimize ‖ ◦√ D − H‖◦√ 2 FDsubject to rankV DV ≤ ρ◦√ √D ∈ EDMNminimize ‖−V ( ◦√ D − H)V ‖◦√ 2 FDsubject to rankV DV ≤ ρ◦√ √D ∈ EDMN(1142)We apply the new convex iteration method for constraining rank. Knownheuristics for solving the problems when compounded with rank minimizationare also explained. We offer a new geometrical proof of a famous resultdiscovered by Eckart & Young in 1936 [85] regarding Euclidean projectionof a point on that generally nonconvex subset of the positive semidefinitecone boundary comprising all positive semidefinite matrices having ranknot exceeding a prescribed bound ρ . We explain how this problem istransformed to a convex optimization for any rank ρ .