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 subproblem. 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 the second-order information of
the objective function. The convergence of the proposed method is proved
under certain assumptions. Numerical results are given to show the
effectiveness of the proposed proximal ADMM.