A Proximal Point Method in Nonreflexive Banach Spaces

Abstract

We propose an inexact version of the proximal point method and study its properties in nonreflexive Banach spaces which are duals of separable Banach spaces, both for the problem of minimizing convex functions and of finding zeroes of maximal monotone operators. By using surjectivity results for enlargements of maximal monotone operators, we prove existence of the iterates in both cases. Then we recover most of the convergence properties known to hold in reflexive and smooth Banach spaces for the convex optimization problem. When dealing with zeroes of monotone operators, our convergence result requests that the regularization parameters go to zero, as is the case for standard (non-proximal) regularization schemes.