An Alternating Direction Method of Multipliers with the BFGS Update for Structured Convex Quadratic Optimization

The alternating direction method of multipliers (ADMM) is an effective method for solving wide fields of convex problems. At each iteration, the classical ADMM solves two subproblems exactly. However, in many applications, it is expensive or impossible to obtain the exact solutions of the subproblems. To overcome the difficulty, some proximal terms are added to the subproblems. This class of methods normally solves the original subproblem approximately, and thus takes more iterations. This fact urges us to consider that a special proximal term can lead to a better result as the classical ADMM. In this paper, we propose a proximal ADMM whose regularized matrix in the proximal term is generated by the BFGS update (or limited memory BFGS) at every iteration. These types of matrices use second-order information of the objective function. The convergence of the proposed method is proved under certain assumptions. Numerical results are presented to show the effectiveness of the proposed proximal ADMM.