Rainer Tichatschke, Universität Trier (Koautor: A. Kaplan)
A general approach for analyzing convergence of proximal-like methods for variational inequalities with set-valued maximal monotone operators is developed.
This approach is devoted to methods coupling successive approximation of the variational inequality with the proximal point algorithm as well as to related methods using regularization on a subspace and/or weak regularization.
The convergence results are proved under mild assumptions with respect to the original variational inequality and admit, in particular, the use of the -enlargement of an operator. Also conditions providing linear convergence are established.
Taking into account the specific structure of non-differentiable terms in energy functionals of several problems in mathematical physics, we analyse the construction of -enlargements for some special operators.
As an application of the general scheme, a proximal-based variant of the elliptic regularization method is considered.