Proximal Methods with Penalization Effects in Banach Spaces

Abstract

In this paper we introduce a boundary coercive condition on the regularizing function of the proximal point method in Banach spaces, which has a penalization effect and is useful for solving certain variational inequality problems. As its finite dimensional counterpart (proximal point method with Bregman distances) it avoids explicit consideration of the constrains, because the feasibility is taken care of by the regularizing function, whose derivative diverges on the boundary of the feasible set. The algorithm is therefore an interior point one. We show that this effect is guaranteed even for an inexact version of the method, namely the hybrid extragradient proximal point method proposed by Solodov and Svaiter for finite dimensional spaces. We give a full convergence analysis, and examples of regularizing functions satisfying the required conditions, for the cases of the feasible set being a closed ball or a polyhedron.