РЕГУЛЯРИЗОВАНИЙ АДАПТИВНИЙ ЕКСТРАПРОКСИМАЛЬНИЙ АЛГОРИТМ ДЛЯ ЗАДАЧІ ПРО РІВНОВАГУ В ПРОСТОРАХ АДАМАРА

• Date: 2020
• Main Authors: Vedel, Ya.I., Denisov, S.V., Semenov, V.V.
• Format: Article
• Language: English
• Published: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Subjects: регуляризація; адаптивність; збіжність; екстрапроксимальний алгоритм; схема Гальперна; задача про рівновагу; псевдомонотонність; простір Адамара
• Online Access: https://jais.net.ua/index.php/files/article/view/487
• Journal Title: Problems of Control and Informatics

One of the intensively developing areas of modern applied nonlinear analysis is the study of equilibrium problems, also known as Ky Fan inequalities, equilibrium programming problems. In the form of an equilibrium problem, one can formulate variational inequalities, mathematical programming problems, and many game theory problems (search of Nash equilibrium). Recently, interest has arisen due to the problems of mathematical biology and machine learning to construct the theory and algorithms for solving mathematical programming problems in Hadamard metric spaces. In this paper, we consider equilibrium problems in Hadamard metric spaces. For an approximate solution of problems, a new iterative regularized adaptive extra-proximal algorithm is proposed and studied. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values ​​at additional points and does not require knowledge of information on of bifunction’s Lipschitz constants. For regularization of basic extra-proximal scheme, the classic Halpern scheme is used. For pseudo-monotone bifunctions of Lipschitz type, the theorem on convergence of sequences generated by the algorithm is proved. The proof is based on the use of the Fejer property of the extra-proximal algorithm with respect to the set of solutions of problem and known results on the convergence of the Halpern scheme. It is shown that the proposed algorithm is applicable to pseudo-monotone variational inequalities in Hilbert spaces.