Convergence of the Operator Extrapolation Method for Variational Inequalities in Banach Spaces

• Published in: Cybernetics and systems analysis Vol. 58; no. 5; pp. 740 - 753
• Main Authors: Semenov, V. V., Denisov, S. V., Sandrakov, G. V., Kharkov, O. S.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2022; Springer; Springer Nature B.V
• Subjects: Adaptive algorithms; Algorithms; Analysis; Artificial Intelligence; Banach spaces; Control; Convergence; Inequalities; Iterative algorithms; Mathematics; Mathematics and Statistics; Methods; Operators (mathematics); Processor Architectures; Software Engineering/Programming and Operating Systems; Systems Theory; Alber generalized projection; gap function; uniformly smooth Banach space; variational inequality; operator extrapolation method; 2-uniformly convex Banach space; weak convergence; monotone operator
• ISSN: 1060-0396; 1573-8337
• DOI: 10.1007/s10559-022-00507-5

New iterative algorithms for solving variational inequalities in uniformly convex Banach spaces are analyzed. The first algorithm is a modification of the forward-reflected-backward method, which uses the Alber generalized projection instead of the metric one. The second algorithm is an adaptive version of the first one, where the monotone step size update rule is used, which does not require knowledge of the Lipschitz constants and linear search procedure. For variational inequalities with monotone Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, theorems on the weak convergence of the methods are proved. Also, for the first algorithm, an efficiency estimate in terms of the gap function is proved.