An Efficient Inexact ABCD Method for Least Squares Semidefinite Programming

Abstract

We consider least squares semidefinite programming (LSSDP) where the primal matrix variable must satisfy given linear equality and inequality constraints, and must also lie in the intersection of the cone of symmetric positive semidefinite matrices and a simple polyhedral set. We propose an inexact accelerated block coordinate descent (ABCD) method for solving LSSDP via its dual, which can be reformulated as a convex composite minimization problem whose objective is the sum of a coupled quadratic function involving four blocks of variables and two separable nonsmooth functions involving only the first and second block, respectively. Our inexact ABCD method has the attractive $O(1/k^2)$ iteration complexity if the subproblems are solved progressively more accurately. The design of our ABCD method relies on recent advances in the symmetric Gauss--Seidel technique for solving a convex minimization problem whose objective is the sum of a multiblock quadratic function and a nonsmooth function involving only th...