A proximal point method for nonsmooth convex optimizationproblems in Banach spaces

Abstract and Applied Analysis

A. N. Iusem³  R. S. Burachik¹  Y. I. Alber² 

[1] Departamento de Matemática, Pontíficia Universidade Católica do Rio de Janeiro, Rua Marqués de São Vicente 225, Rio de Janeiro, RJ CEP 22453-030, Brazil, puc-rio.br;Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ CEP 22460-320, Brazil, impa.br;Department of Mathematics, The Technion-Israel Institute of Technology, Haifa 32000, Israel, technion.ac.il;Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ CEP 22460-320, Brazil, impa.br

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

Others  :  1361321
DOI  :  10.1155/S1085337597000298

实施日期：1996-08-21，发布日期：1996-08-21

【 摘 要 】

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 estimatedepends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includesBanach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators inBanach spaces.

【 授权许可】

   
Copyright © 1997 Hindawi Publishing Corporation 1997