Generalized Mirror Prox Algorithm for Monotone Variational Inequalities: Universality and Inexact Oracle

Fedor Stonyakin (), Alexander Gasnikov (), Pavel Dvurechensky (), Alexander Titov () and Mohammad Alkousa ()
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Fedor Stonyakin: V.I. Vernadsky Crimean Federal University
Alexander Gasnikov: Moscow Institute of Physics and Technology
Pavel Dvurechensky: Weierstrass Institute for Applied Analysis and Stochastics
Alexander Titov: Moscow Institute of Physics and Technology
Mohammad Alkousa: Moscow Institute of Physics and Technology

Journal of Optimization Theory and Applications, 2022, vol. 194, issue 3, No 10, 988-1013

Abstract: Abstract We introduce an inexact oracle model for variational inequalities with monotone operators, propose a numerical method that solves such variational inequalities, and analyze its convergence rate. As a particular case, we consider variational inequalities with Hölder-continuous operator and show that our algorithm is universal. This means that, without knowing the Hölder exponent and Hölder constant, the algorithm has the least possible in the worst-case sense complexity for this class of variational inequalities. We also consider the case of variational inequalities with a strongly monotone operator and generalize the algorithm for variational inequalities with inexact oracle and our universal method for this class of problems. Finally, we show how our method can be applied to convex–concave saddle point problems with Hölder-continuous partial subgradients.

Keywords: Variational inequality; monotone operator; Hölder continuity; Inexact oracle; Complexity estimate; 65K15; 90C33; 90C06; 68Q25; 65Y20; 68W40; 58E35 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-022-02062-7

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