Improving an ADMM-like Splitting Method via Positive-Indefinite Proximal Regularization for Three-Block Separable Convex Minimization

Abstract. The augmented Lagrangian method (ALM) is fundamental for solving convex minimization models with linear constraints. When the objective function is separable such that it can be represented as the sum of more than one function without coupled variables, various splitting versions of the ALM have been well studied in the literature such as the alternating direction method of multipliers (ADMM) for the two-block separable case. It is known that the multi-block separable case where the objective function is the sum of more than two functions is more complicated than the two-block case; and it requires particular discussions on its theoretical convergence and algorithmic design. The ADMM-like splitting method in [12] allows some of its subproblems to be solved in parallel; while these subproblems should be regularized by some positive-definite proximal terms to ensure the convergence. In this paper, we further study this partially-parallel ADMM-like splitting method for a three-block separable convex minimization model and discuss how to relax the positive-definite proximal regularization terms in its subproblems. Inspired by our recent work in [10] for the ALM and ADMM, we show that it is not necessary to regularize the splitted subproblems by positive-definite proximal regularization for the method in [12]. Indeed, the proximal terms can be induced by a positive-indefiniteness proximal matrix. Thus, larger step sizes for solving these subproblems are allowed. This is an important feature to embark efficient algorithms with faster convergence. The convergence and worst-case convergence rate of the improved version of the ADMM-like splitting method in [12] with a less restricted proximal parameter are proved.