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Convergence of the Operator Extrapolation Method for Variational Inequalities in Banach Spaces*

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Cybernetics and Systems Analysis Aims and scope

Abstract

New iterative algorithms for solving variational inequalities in uniformly convex Banach spaces are analyzed. The first algorithm is a modification of the forward-reflected-backward method, which uses the Alber generalized projection instead of the metric one. The second algorithm is an adaptive version of the first one, where the monotone step size update rule is used, which does not require knowledge of the Lipschitz constants and linear search procedure. For variational inequalities with monotone Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, theorems on the weak convergence of the methods are proved. Also, for the first algorithm, an efficiency estimate in terms of the gap function is proved.

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Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    V. V. Semenov, S. V. Denisov, G. V. Sandrakov & O. S. Kharkov

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  1. V. V. Semenov
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  2. S. V. Denisov
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  3. G. V. Sandrakov
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  4. O. S. Kharkov
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Correspondence to V. V. Semenov.

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*The study was financially supported by the National Academy of Sciences of Ukraine (project “New Methods for the Correctness Analysis and Solution of Discrete Optimization Problems, Variational Inequalities, and their Application,” SR 0119U101608) and Ministry of Education and Science of Ukraine (project “Computational Algorithms and Optimization for Artificial Intelligence, Medicine, and Defense,” SR 0122U002026).

Translated from Kibernetyka ta Systemnyi Analiz, No. 5, September–October, 2022, pp. 79–93.

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Semenov, V.V., Denisov, S.V., Sandrakov, G.V. et al. Convergence of the Operator Extrapolation Method for Variational Inequalities in Banach Spaces*. Cybern Syst Anal 58, 740–753 (2022). https://doi.org/10.1007/s10559-022-00507-5

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