Modified alternating direction method of multipliers for convex quadratic semidefinite programming

The dual form of convex quadratic semidefinite programming (CQSDP) problem, with nonnegative constraints on the matrix variable, is a 4-block convex optimization problem. It is known that, the directly extended 4-block alternating direction method of multipliers (ADMM) is very efficient to solve this dual, but its convergence are not guaranteed. In this paper, we reformulate it as a 3-block convex programming by introducing an extra variable, and propose a new modified ADMM to solve it. At each iteration, the proposed method does not have to work out the sub-problem with the primal variable, compared with the existing ADMM methods. This confirms that at least our methods require less computation than the existing ADMM in one iteration. Under satisfying a condition on the penalty parameter, the convergence of modified ADMM to a KKT point is proved by using a fixed-point argument. Numerical experiments on the various classes of CQSDP problems (including least squares semidefinite programming (LSSDP)) show that, our proposed algorithm performs comparable or slightly better than the directly extended 4-block ADMM with unit step-length.