Abstract and Applied Analysis

Abstract and Applied Analysis / 1997 / Article

Open Access

Volume 2 |Article ID 614871 | 24 pages | https://doi.org/10.1155/S1085337597000298

A proximal point method for nonsmooth convex optimization problems in Banach spaces

Received21 Aug 1996

Abstract

In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces.

Copyright © 1997 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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