An explicit algorithm for solving monotone variational inequalities

Vuong, Phan Tu, Gibali, Aviv and Thong, Dong Viet (2022) An explicit algorithm for solving monotone variational inequalities. Applied Numerical Mathematics, 171, 408-425. (doi:10.1016/j.apnum.2021.09.013).

Record type: Article

Abstract

In this paper, we are interested in the generalized variational inequality problem in real Hilbert spaces. We propose an explicit proximal method which requires only one proximal step and one mapping evaluation per iteration and also uses an adaptive step-size rule that enables to avoid the prior knowledge of the Lipschitz constant of the involved mapping. Weak convergence of the proposed scheme is established under standard assumptions. Under strong monotonicity, we present the R-linear convergence rate of our new method. Intensive numerical experiments illustrate the advantages and the applicability of our scheme. Moreover, our work generalizes theoretically several recent results in this field.

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Accepted/In Press date: 20 September 2021

e-pub ahead of print date: 23 September 2021

Published date: January 2022

Additional Information: Funding Information: The authors thank Prof. Hedy Attouch, Juan Peypouquet and Yura Malisky for independent fruitful discussions and insights providing while preparing this work. They are grateful to three anonymous referees for their carefully reading and comments, which help to improve the presentation of the paper. This research is funded by National Economics University , Hanoi, Vietnam. Publisher Copyright: © 2021 IMACS

Keywords: Gene expression, Linear convergence, Monotonicity, Tomography, Variational inequality, Weak convergence