A General Approach to Convergence Properties of Some Methods for Nonsmooth Convex Optimization
- Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109 (United States)
- School of Mathematics, University of New South Wales, Sydney, NSW 2052 (Australia)
Based on the notion of the {epsilon} -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, Lemarechal, and Sagastizabal, (ii) some algorithms proposed by Correa and Lemarechal, 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 {l_brace} parallel x{sub k} parallel {r_brace} and {l_brace}f(x{sub k}) {r_brace} when {l_brace}x{sub k} {r_brace} is unbounded and {l_brace}f(x{sub k}) {r_brace} is bounded for the non-smooth minimization methods (i), (ii), and (iii)
- OSTI ID:
- 21067565
- Journal Information:
- Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 2 Vol. 38; ISSN 0095-4616
- Country of Publication:
- United States
- Language:
- English
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