ЗБІЖНІСТЬ ЕКСТРАГРАДІЄНТНОГО АЛГОРИТМУ З МОНОТОННИМ РЕГУЛЮВАННЯМ КРОКУ ДЛЯ ВАРІАЦІЙНИХ НЕРІВНОСТЕЙ ТА ОПЕРАТОРНИХ РІВНЯНЬ

• Date: 2025
• Main Authors: Denisov, S.V., Nomirovskii, D.A., Rublyov, B.V., Semenov, V.V.
• Format: Article
• Language: English
• Published: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Subjects: слабка збіжність; екстраградієнтний алгоритм Корпелевич; варіаційна нерівність; операторне рівняння; псевдомонотонність; квазінерозтягуючий оператор; нерухома точка
• Online Access: https://jais.net.ua/index.php/files/article/view/649
• Journal Title: Problems of Control and Informatics

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.