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An Efficient Inexact ABCD Method for Least Squares Semidefinite Programming

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 non-smooth 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 multi-block quadratic function and a non-smooth function involving only the first block. Extensive numerical experiments on various classes of over 600 large scale LSSDP problems demonstrate that our proposed ABCD method not only can solve the problems to high accuracy, but it is also far more efficient than (a) the well known BCD (block coordinate descent) method, (b) the eARBCG (an enhanced version of the accelerated randomized block coordinate gradient) method, and (c) the APG (accelerated proximal gradient) method.