Proximal alternating direction-based contraction methods for separable linearly constrained convex optimization

Abstract

Alternating direction method (ADM) has been well studied in the context of linearly constrained convex programming problems. Recently, because of its significant efficiency and easy implementation in novel applications, ADM is extended to the case where the number of separable parts is a finite number. The algorithmic framework of the extended method consists of two phases. At each iteration, it first produces a trial point by using the usual alternating direction scheme, and then the next iterate is updated by using a distance-descent direction offered by the trial point. The generated sequence approaches the solution set monotonically in the Fejer sense, and the method is called alternating direction-based contraction (ADBC) method. In this paper, in order to simplify the subproblems in the first phase, we add a proximal term to the objective function of the minimization subproblems. The resulted algorithm is called proximal alternating direction-based contraction (PADBC) methods. In addition, we present different linearized versions of the PADBC methods which substantially broaden the applicable scope of the ADBC method. All the presented algorithms are guided by a general framework of the contraction methods for monotone variational inequalities, and thus, the convergence follows directly.