Abstract

Many algorithms for solving the problem of finding zeroes of a sum of two maximal monotone operators $T1$ and $T2$, have regularized subproblems of the kind $0 \in T1(x) + T2(x) + \partial D(x)$, where $D $is a convex function. We develop an unified analysis for existence of solutions of these subproblems, through the introduction of the concept of convex regularization, which includes several well-known cases in the literature. Finally, we establish conditions, either on D or on the operators, which assure solvability of the subproblems.