Convergence of Halpern's Iteration Method with Applications in Optimization. (18th November 2021)

• Record Type: Journal Article
• Authors: Qi, Huiqiang; Xu, Hong-Kun
• Is Part Of: Numerical functional analysis and optimization. Volume 42:Number 15(2021)
• Page Start: 1839
• Page End: 1854
• Publication Date: 2021-11-18
• Subjects: Halpern iteration -- inverse strongly monotone -- monotone inclusion -- nonexpansive mapping -- projection -- variational inequality; 47H09 -- 47J20 -- 47J26; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.2001826
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: Halpern's iteration method, discovered by Halpern in 1967, is an iterative algorithm for finding fixed points of a nonexpansive mapping in Hilbert and Banach spaces. Since many optimization problems can be cast into fixed point problems of nonexpansive mappings, Halpern's method plays an important role in optimization methods. This paper discusses recent advances in convergence and rate of convergence results of Halpern's method, and applications in optimization problems, including variational inequalities, monotone inclusions, Douglas-Rachford splitting method, and minimax problems.