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Primal-dual subgradient methods for convex problems


In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primal-dual 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 (however, we always assume the uniform boundedness of subgradients). 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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Corresponding author

Correspondence to Yurii Nesterov.

Additional information

Dedicated to B. T. Polyak on the occasion of his 70th birthday

The research results presented in this paper have been supported by a grant “Action de recherche concertè ARC 04/09-315” from the “Direction de la recherche scientifique - Communautè française de Belgique”. The scientific responsibility rests with the author.

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Nesterov, Y. Primal-dual subgradient methods for convex problems. Math. Program. 120, 221–259 (2009).

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  • Convex optimization
  • Subgradient methods
  • Non-smooth optimization
  • Minimax problems
  • Saddle points
  • Variational inequalities
  • Stochastic optimization
  • Black-box methods
  • Lower complexity bounds

Mathematics Subject Classification (2000)

  • 90C25
  • 90C47
  • 68Q25