A Class of Variable Metric Decomposition Methods for Monotone Variational Inclusions

We extend the general decomposition scheme of M. V. Solodov [Optimization Methods and Software 19 (2004) 557--575], which is based on the hybrid inexact proximal point method of M. V. Solodov and B. F. Svaiter [Numerical Functional Analysis and Optimization 22 (2001) 1013--1035], to allow the use of variable metric in subproblems, along the lines described in a previous paper of the authors [SIAM Journal on Optimization 19 (2008) 240--260]. We show that the new general scheme includes as special cases the splitting method for composite mappings [see T. Pennanen, Numerical Functional Analysis and Optimization 23 (2002) 875--890] and the proximal alternating directions method [see J. Eckstein, Optimization Methods and Software 4 (1994) 75--83, and B. He, L. Z. Liao, D. Han and H. Yang, Mathematical Programming 92 (2002) 103--118] (in addition to the decomposition methods of X. Chen and M. Teboulle [Mathematical Programming 64 (1994) 81--101] and P. Tseng [SIAM Journal on Optimization 7 (1997) 951--965] that were already covered in the above-mentioned article by M. V. Solodov [Optimization Methods and Software 19 (2004) 557--575]). 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.