Abstract and Applied Analysis
Volume 2 (1997), Issue 1-2, Pages 97-120
doi:10.1155/S1085337597000298
A proximal point method for nonsmooth convex optimization
problems in Banach spaces
1Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro CEP 22460-320, RJ, Brazil
2Departamento de Matemática, Pontíficia Universidade Católica do Rio de Janeiro, Rua Marqués de São Vicente 225, Rio de Janeiro CEP 22453-030, RJ, Brazil
3Department of Mathematics, The Technion-Israel Institute of Technology, Haifa 32000, Israel
Received 21 August 1996
Copyright © 1997 Y. I. Alber et al. 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.
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.