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Mirror Descent and Constrained Online Optimization Problems

  • Alexander A. TitovEmail author
  • Fedor S. Stonyakin
  • Alexander V. Gasnikov
  • Mohammad S. Alkousa
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)

Abstract

We consider the following class of online optimization problems with functional constraints. Assume, that a finite set of convex Lipschitz-continuous non-smooth functionals are given on a closed set of n-dimensional vector space. The problem is to minimize the arithmetic mean of functionals with a convex Lipschitz-continuous non-smooth constraint. In addition, it is allowed to calculate the (sub)gradient of each functional only once. Using some recently proposed adaptive methods of Mirror Descent the method is suggested to solve the mentioned constrained online optimization problem with an optimal estimate of accuracy. For the corresponding non-Euclidean prox-structure, the case of a set of n-dimensional vectors lying on the standard n-dimensional simplex is considered.

Keywords

Online convex optimization Non-smooth constrained optimization Adaptive mirror descent Non-euclidean prox-structure Unit simplex 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexander A. Titov
    • 1
    Email author
  • Fedor S. Stonyakin
    • 1
    • 2
  • Alexander V. Gasnikov
    • 1
    • 3
  • Mohammad S. Alkousa
    • 1
  1. 1.Moscow Institute of Physics and TechnologiesMoscowRussia
  2. 2.V. I. Vernadsky Crimean Federal UniversitySimferopolRussia
  3. 3.Adyghe State University, Caucasus Mathematical CenterMaykopRussia

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