SPLITTING METHODS FOR FINDING ZEROES OF SUMS OF MAXIMAL MONOTONE OPERATORS IN BANACH SPACES

We introduce a general scheme for finding zeroes of the sum of two 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 in 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. Thus, the resulting procedure is a splitting method, which solves subproblems involving only one of the two operators. Similarly to other methods designed for Banach spaces, auxiliary functions, giving rise to Breman distances and Bregman projections, are used in both the proximal-like step and in the projection step of the scheme. A full convergence analysis is presented.