A variational inequalities and operator equations in an infinite dimensional Hilbert space with additional conditions for the type of inclusion in the set of fixed points of a given operator are considered. For an approximate solution of the problems, a novel iterative algorithm that is a superposition of a modified Korpelevich extragradient algorithm with monotone step-size strategy that does not require knowledge of the Lipschitz constant of operator, and the Krasnoselskii–Mann scheme for approximating fixed points, is proposed. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not perform additional calculations for the operator values and the projections mapping. The algorithm was investigated using the theory of iterative processes of the Fejer type. The weak convergence of the algorithm for problems with pseudo-monotone, Lipschitz-continuous, and sequentially weakly continuous operators and quasi-nonexpansive operators, which specify additional conditions is proved. Previously, similar results on weak convergence were known only for variational inequalities with monotone, Lipschitz-continuous operators and with nonexpansive operators, which specify additional conditions.

Chapter 7 Modified Extragradient Algorithms for Variational Inequalities Vladimir V. SEMENOV, Vladimir V. SEMENOV Taras Shevchenko National University of Kyiv, UkraineSearch for more papers by this authorSergey V. DENISOV, Sergey V. DENISOV Taras Shevchenko National University of Kyiv, UkraineSearch for more papers by this author Vladimir V. SEMENOV, Vladimir V. SEMENOV Taras Shevchenko National University of Kyiv, UkraineSearch for more papers by this authorSergey V. DENISOV, Sergey V. DENISOV Taras Shevchenko National University of Kyiv, UkraineSearch for more papers by this author Dmitri Koroliouk, Dmitri KorolioukSearch for more papers by this authorSergiy Lyashko, Sergiy LyashkoSearch for more papers by this authorNikolaos Limnios, Nikolaos LimniosSearch for more papers by this author Book Author(s):Dmitri Koroliouk, Dmitri KorolioukSearch for more papers by this authorSergiy Lyashko, Sergiy LyashkoSearch for more papers by this authorNikolaos Limnios, Nikolaos LimniosSearch for more papers by this author First published: 19 April 2024 https://doi.org/10.1002/9781394284344.ch7 AboutPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onEmailFacebookTwitterLinkedInRedditWechat Summary This chapter discusses different variants of the modified extragradient algorithm for solving variational inequalities with monotone operators acting in the Hilbert spaces. Algorithms use the variable step sizes which are updated over each iteration by some cheap computations. These step sizes are found without the prior knowledge of the Lipschitz constant of an operator. Projective methods for variational inequalities originate from the works of A. A. Goldstein and E. S. Levitin and B. T. Polyak, where they independently proposed a gradient projection method for a conditional optimization problem. The chapter investigates the variant of the algorithm for finding a solution of variational inequality which is also a fixed point of some provided mapping. It describes variants of the method for variational inequalities and operator equations with a priori information about the solution given as a set of fixed points of the quasi-non-expansive operator. References Antipin , A.S. ( 1976 ). On a convex programming method using a symmetrical modification of the Lagrange function . Matecon. , 12 ( 6 ), 1164 – 1173 . Google Scholar Aubin , J.P. and Ekeland , I. 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