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Primal-dual subgradient methods for convex problems


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    In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primaldual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. The proposed schemes differ from the classical approaches (divergent series methods, mirror descent methods) by presence of two control sequences. The first sequence is responsible for aggregating the support functions in the dual space, and the second one establishes a dynamically updated scale between the primal and dual spaces. This additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set. We present the variants of subgradient schemes for nonsmooth convex minimization, minimax problems, saddle point problems, variational inequalities, and stochastic optimization. In all situations our methods are proved to be optimal from the view point of worst-case black-box lower complexity bounds.

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    Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2005067.

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    Date of creation: 00 Oct 2005
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    Handle: RePEc:cor:louvco:2005067

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    Keywords: convex optimization; subgradient methods; non-smooth optimization; minimax problems; saddle points; variational inequalities; stochastic optimization; black-box methods; lower complexity bounds;


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    1. NESTEROV, Yu, 2003. "Dual extrapolation and its applications for solving variational inequalities and related problems," CORE Discussion Papers 2003068, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. repec:fth:louvco:2000/13 is not listed on IDEAS
    3. Nesterov, Y. & Vial, J.-P., 2000. "Confidence Level Solutions for Stochastic Programming," Papers 2000.05, Ecole des Hautes Etudes Commerciales, Universite de Geneve-.
    4. NESTEROV, Yu & VIAL, Jean-Philippe, 2000. "Confidence level solutions for stochastic programming," CORE Discussion Papers 2000013, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:
    1. NESTEROV, Yu., 2005. "Minimizing functions with bounded variation of subgradients," CORE Discussion Papers 2005079, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).


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