WIAS Preprint No. 2709, (2020)

Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model


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

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25 65K15


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




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, Bregman 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 conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities.

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