A Class of Variable Metric Decomposition Methods for Monotone Variational Inclusions

We extend the general decomposition scheme of [32], which is based on the hybrid inexact proximal point method of [38], to allow the use of variable metric in subproblems, along the lines of [23]. We show that the new general scheme includes as special cases the splitting method for composite mappings [25] and the proximal alternating directions method [13, 17] (in addition to the decomposition methods of [10, 42] that were already covered by [32]). Apart from giving a unified insight into the decomposition methods in question and openning the possibility of using variable metric, which is a computationally important issue, this development also provides linear rate of convergence results not previously available for splitting of composite mappings and for the proximal alternating directions methods.