АДАПТИВНИЙ ЕКСТРАПРОКСИМАЛЬНИЙ АЛГОРИТМ ДЛЯ ЗАДАЧІ ПРО РІВНОВАГУ В ПРОСТОРАХ АДАМАРА

• Datum: 2020
• Hauptverfasser: Vedel, Ya.I., Golubeva, E.N., Semenov, V.V., Chabak, L.M.
• Format: Artikel
• Sprache: English
• Veröffentlicht: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Schlagworte: простір Адамара; задача про рівновагу; псевдомонотонність; екстрапроксимальний алгоритм; адаптивність; збіжність
• Online Zugang: https://jais.net.ua/index.php/files/article/view/476
• Назва журналу: Problems of Control and Informatics

One of the most popular areas of modern applied nonlinear analysis is the study of equilibrium problems (Ky Fan inequalities, equilibrium programming problems). In the form of an equilibrium problem, one can formulate mathematical programming problems, vector optimization problems, variational inequalities, and many game theory problems. The classical formulation of the equilibrium problem first appeared in the works of H. Nikaido and K. Isoda, and the first general proximal algorithms for solving equilibrium problems were proposed by A.S. Antipin. 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. Another strong motivation for studying these problems is the ability to write down some nonconvex problems in the form of convex (more precisely, geodesically convex) in a space with a specially selected metric. In this paper, we consider general equilibrium problems in Hadamard metric spaces. For an approximate solution of problems, a new iterative adaptive extra-proximal algorithm is proposed and studied. At each step of the algorithm, sequential minimization of two special strongly convex functions should be done. 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 pseudo-monotone bifunctions of Lipschitz type, weakly upper semicontinuous in the first variable, convex and lower semicontinuous in the second variable, the theorem on weak convergence of sequences generated by the algorithm is proved. The proof is based on the use of the Fejer property of the algorithm with respect to the set of solutions of equilibrium problem. It is shown that the proposed algorithm is applicable to variational inequalities with Lipschitzcontinuous, sequentially weakly continuous and pseudomonotone operators acting in Hilbert spaces.