v2007.09.13 - 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 (724),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 (1076)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 +(1070)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 to a givenmatrix in some sense. We apply convex iteration and also explainknown heuristics for solving the problems when compounded with rankminimization: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(1112)We offer a new geometrical proof of a famous result discovered byEckart & Young in 1936 [84] regarding Euclidean projection of a point onthat generally nonconvex subset of the positive semidefinite cone boundarycomprising all positive semidefinite matrices having rank not exceeding aprescribed bound ρ . We explain how this problem is transformed to aconvex optimization for any rank ρ .