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

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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

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

First available in Project Euclid: 7 April 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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