A note on the existence of zeroes of convexly regularized sums of maximal monotone operators

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 ∈ T1(x)+T2(x)+∂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.  2003 Elsevier Science (USA). All rights reserved.