Iteration-Complexity of Block-Decomposition Algorithms and the Alternating Direction Method of Multipliers

In this paper, we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point-to-set maximal monotone operator with a separable two-block structure and introduce a framework of block-decomposition prox-type algorithms for solving it which allows for each one of the single-block proximal subproblems to be solved in an approximate sense. Moreover, by showing that any method in this framework is also a special instance of the hybrid proximal extragradient (HPE) method introduced by Solodov and Svaiter, we derive corresponding convergence rate results. We also describe some instances of the framework based on specific and inexpensive schemes for solving the single-block proximal subproblems. Finally, we consider some applications of our methodology to establish for the first time (i) the iteration-complexity of an algorithm for finding a zero of the sum of two arbitrary maximal monotone operators and, as a consequence, the ergodic iteration-complexity of the Douglas--Rachford splitting method and (ii) the ergodic iteration-complexity of the classical alternating direction method of multipliers for a class of linearly constrained convex programming problems with proper closed convex objective functions.