WIAS Preprint No. 2679, (2020)

Inexact model: A framework for optimization and variational inequalities



Authors

  • Stonyakin, Fedor
  • Gasnikov, Alexander
    ORCID: 0000-0003-1201-2343
  • Tyurin, Alexander
  • Pasechnyuk, Dmitry
  • Agafonov, Artem
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Dvinskikh, Darina
  • Piskunova, Victorya

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25

2008 Physics and Astronomy Classification Scheme

  • 65K15

Keywords

  • Convex optimization, composite optimization, proximal method, level-set method, variational inequality, universal method, mirror prox, acceleration, relative smoothness

DOI

10.20347/WIAS.PREPRINT.2679

Abstract

In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities.

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