Abstract and Applied Analysis

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

Y. I. Alber, R. S. Burachik, and A. N. Iusem

Full-text: Open access

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.

Article information

Source
Abstr. Appl. Anal., Volume 2, Number 1-2 (1997), 97-120.

Dates
First available in Project Euclid: 7 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1049737245

Digital Object Identifier
doi:10.1155/S1085337597000298

Mathematical Reviews number (MathSciNet)
MR1604165

Zentralblatt MATH identifier
0947.90091

Subjects
Primary: 90C25: Convex programming
Secondary: 49D45 49D37

Keywords
Proximal point algorithm Banach spaces duality mappings nonsmooth and convex functionals subdifferentials moduli of convexity and smoothness of Banach spaces generalized projection operators Lyapunov functionals convergence stability estimates of convergence rate

Citation

Alber, Y. I.; Burachik, R. S.; Iusem, A. N. A proximal point method for nonsmooth convex optimization problems in Banach spaces. Abstr. Appl. Anal. 2 (1997), no. 1-2, 97--120. doi:10.1155/S1085337597000298. https://projecteuclid.org/euclid.aaa/1049737245


Export citation