A General Approach to Convergence Properties of Some Methods for Nonsmooth Convex Optimization |
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| Authors: | J R Birge L Qi Z Wei |
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| Institution: | (1) Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA , US;(2) School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia , AU |
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| 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).
Accepted 15 October 1996 |
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| Keywords: | , Nonsmooth convex minimization, Global convergence, Convergence rate, AMS Classification, 90C25, 90C30, 90C33, |
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