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1997 A proximal point method for nonsmooth convex optimization problems in Banach spaces
Y. I. Alber, R. S. Burachik, A. N. Iusem
Abstr. Appl. Anal. 2(1-2): 97-120 (1997). DOI: 10.1155/S1085337597000298


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.


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Y. I. Alber. R. S. Burachik. A. N. Iusem. "A proximal point method for nonsmooth convex optimization problems in Banach spaces." Abstr. Appl. Anal. 2 (1-2) 97 - 120, 1997.


Published: 1997
First available in Project Euclid: 7 April 2003

zbMATH: 0947.90091
MathSciNet: MR1604165
Digital Object Identifier: 10.1155/S1085337597000298

Primary: 90C25
Secondary: 49D37 , 49D45

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

Rights: Copyright © 1997 Hindawi

Vol.2 • No. 1-2 • 1997
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