  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.
