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Approximate Level Method for Nonsmooth Convex Minimization

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Abstract

In this paper, we propose and analyse an approximate variant of the level method of Lemaréchal, Nemirovskii and Nesterov for minimizing nonsmooth convex functions. The main per-iteration work of the level method is spent on (i) minimizing a piecewise-linear model of the objective function and (ii) projecting onto the intersection of the feasible region and a level set of the model function. We show that, by replacing exact computations in both cases by approximate computations, in relative scale, the theoretical iteration complexity increases only by a small factor which depends on the approximation level and reduces to one in the exact case.

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Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh, EH9 3JZ, UK

    Peter Richtárik

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  1. Peter Richtárik
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Correspondence to Peter Richtárik.

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Communicated by Marcin Studniarski.

The research results presented in this paper have been supported by a grant “Action de recherche concertée ARC 04/09-315” from the “Direction de la recherche scientifique—Communauté française de Belgique”. This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the author(s).

The author wishes to thank Yurii Nesterov for enlightening discussions and suggestions that greatly helped to improve the paper, and to Robert Chares and two anonymous referees for carefully reading the manuscript.

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Richtárik, P. Approximate Level Method for Nonsmooth Convex Minimization. J Optim Theory Appl 152, 334–350 (2012). https://doi.org/10.1007/s10957-011-9908-1

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  • DOI: https://doi.org/10.1007/s10957-011-9908-1

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