Preprint A233/2003
A strongly convergent Hybrid Proximal method in Banach spaces

B. F. Svaiter | Gárciga Otero , Rolando

**Keywords: **
proximal point method | relative error | inexact solutions | strong convergence

This paper is devoted to the study of strong convergence in
inexact
proximal like methods for finding zeroes of maximal monotone operators
in Banach spaces. Convergence properties of proximal point methods in Banach spaces
can be summarized as follows: if the operator have zeroes then
the sequence of iterates is bounded and
all its weak accumulation points are solutions.
Whether or not the whole sequence
converges weakly to a solution and which is the relation of the weak limit with
the initial iterate are key questions. We present a hybrid proximal
Bregman projection method, allowing for inexact solutions of the proximal
subproblems, that guarantees strong convergence of the sequence to the closest
solution, in the sense of the Bregman distance, to the initial iterate.