Iteration-complexity analysis of a generalized alternating direction method of multipliers

摘要

This paper analyzes the iteration-complexity of a generalized alternating direction method of multipliers (G-ADMM) for solving separable linearly constrained convex optimization problems. This ADMM variant, first proposed by Bertsekas and Eckstein, introduces a relaxation parameter into the second ADMM subproblem in order to improve its computational performance. It is shown that, for a given tolerance epsilon0, the G-ADMM with (0,2) provides, in at most O(1/epsilon 2) iterations, an approximate solution of the Lagrangian system associated to the optimization problem under consideration. It is further demonstrated that, in at most O(1/epsilon) iterations, an approximate solution of the Lagrangian system can be obtained by means of an ergodic sequence associated to a sequence generated by the G-ADMM with (0,2]. Our approach consists of interpreting the G-ADMM as an instance of a hybrid proximal extragradient framework with some special properties. Some preliminary numerical experiments are reported in order to confirm that the use of 1 can lead to a better numerical performance than =1 (which corresponds to the standard ADMM).