Preprint A747/2014
Spliting methods for finding zeroes of sums of maximal monotone operators in Banach spaces
Alfredo Noel Iusem | Benar Fux Svaiter
Keywords: Splitting methods | projection methods | reflexive Banach spaces | Douglas-Rachford method

We introduce a general scheme for finding zeroes of sums of maximal monotone operators in a reflexive Banach space X. It generates a sequence in the product space X x X*, where X* is the dual of X. It is essentially a projection method, in the sense that in each iteration a hyperplane is constructed, separating the current iterate from a generalized solution set, whose projection onto X is indeed the solution set of the problem, and then the next iterate is taken as the projection of the current one onto this separating hyperplane. In order to construct such hyperplane, two proximal-like steps are taken from the current iterate, each one using only one of the two maximal monotone operators. Similarly to other methods designed for Banach spaces, auxiliary functions, giving rise to Bregman distances and Bregman projectios, are used in both the proximal-like step and in the projection step of the scheme. A full convergence analysis is presented.