Adaptive extraproximal algorithm for the equilibrium problem in the hadamard spaces

被引：0

作者：

Vedel Ya.I. ^{[1
]}

Golubeva E.N. ^{[1
]}

Semenov V.V. ^{[1
]}

Chabak L.M. ^{[2
]}

机构：

[1] State University of Infrastructure and Technology, Kiev

来源：

Journal of Automation and Information Sciences | 2020年 / 52卷 / 08期

关键词：

Adaptivity; Convergence; Equilibrium problem; Extraproximal algorithm; Hadamard space; Pseudomonotonicity;

D O I：

10.1615/JAUTOMATINFSCIEN.V52.I8.40

中图分类号：

学科分类号：

摘要：

One of the most popular directions of modern applied nonlinear analysis is the study of equilibrium problems (Ky Fan inequalities, equilibrium programming problems). It is possible to formulate problems of mathematical programming problems, vector optimization problems, variational inequalities, and many game theory problems in the form of an equilibrium problem. 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 appeared due to the problems of mathematical biology and machine learning to construct the theory and algorithms for solving mathematical programming problems in the 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 the Hadamard metric spaces. For an approximate problem solving a new iterative adaptive extraproximal algorithm is proposed and studied. At every 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 the information about the Lipschitz constants of bifunctions. For pseudo-monotone bifunctions of Lipschitz type weakly upper semicontinuous relative to the first variable, convex and lower semicontinuous relative to the second variable, the theorem about weak convergence of sequences generated by the algorithm is proved. The proof is based on the use of the Fejer property of the algorithm for the set of solutions of the equilibrium problem. It is shown that the proposed algorithm is applicable to variational inequalities with the Lipschitz-continuous, sequentially weakly continuous and pseudomonotone operators acting in the Hilbert spaces. © 2020 by Begell House Inc.

引用

页码：46 / 58

页数：12

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