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A General Approach to Convergence Properties of Some Methods for Nonsmooth Convex Optimization

A General Approach to Convergence Properties of Some Methods for Nonsmooth Convex Optimization Abstract. Based on the notion of the ε -subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemaréchal, and Sagastizábal, (ii) some algorithms proposed by Correa and Lemaréchal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar—Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of {||x k || } and {f(x k ) } when {x k } is unbounded and {f(x k ) } is bounded for the non\-smooth minimization methods (i), (ii), and (iii). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A General Approach to Convergence Properties of Some Methods for Nonsmooth Convex Optimization

Applied Mathematics and Optimization , Volume 38 (2): 18 – Oct 1, 1998

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References (12)

Publisher
Springer Journals
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s002459900086
Publisher site
See Article on Publisher Site

Abstract

Abstract. Based on the notion of the ε -subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemaréchal, and Sagastizábal, (ii) some algorithms proposed by Correa and Lemaréchal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar—Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of {||x k || } and {f(x k ) } when {x k } is unbounded and {f(x k ) } is bounded for the non\-smooth minimization methods (i), (ii), and (iii).

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 1998

Keywords: Key words. Nonsmooth convex minimization, Global convergence, Convergence rate. AMS Classification. 90C25, 90C30, 90C33.

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