v2010.10.26 - Convex Optimization

32 CHAPTER 1. OVERVIEWto a given matrix H :minimize ‖−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(1310)We apply a convex iteration method for constraining rank. Known heuristicsfor rank minimization are also explained. We offer new geometrical proof, ofa famous discovery by Eckart & Young in 1936 [129], with particular regardto Euclidean projection of a point on that generally nonconvex subset ofthe positive semidefinite cone boundary comprising all semidefinite matriceshaving rank not exceeding a prescribed bound ρ . We explain how thisproblem is transformed to a convex optimization for any rank ρ .appendicesToolboxes are provided so as to be more self-contained:linear algebra (appendix A is primarily concerned with properstatements of semidefiniteness for square matrices),simple matrices (dyad, doublet, elementary, Householder, Schoenberg,orthogonal, etcetera, in appendix B),collection of known analytical solutions to some important optimizationproblems (appendix C),matrix calculus remains somewhat unsystematized when comparedto ordinary calculus (appendix D concerns matrix-valued functions,matrix differentiation and directional derivatives, Taylor series, andtables of first- and second-order gradients and matrix derivatives),