A splitting method for finding the resolvent of the sum of two maximal monotone operators

• Published in: Optimization Vol. 71; no. 7; pp. 1863 - 1882
• Main Author: Matsushita, Shin-ya
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.07.2022; Taylor & Francis LLC
• Subjects: Computation; Convergence; fixed point; Hilbert space; Mapping; maximal monotone; nonexpansive; Operators; Optimization; Ratios; resolvent; Splitting; Splitting method
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2020.1839068

This paper considers the problem of finding the resolvent of the sum of two maximal monotone operators. Such a problem arises frequently in practice, but it seems that computation of the solution of the problem is not necessarily easy. It is assumed that both the resolvents of two maximal monotone operators can be easily computed. This enables us to consider the case in which a solution to the problem cannot be computed easily. This paper introduces a new mapping, which satisfies the nonexpansivity property, from the individual resolvents of two maximal monotone operators and investigates some of its properties. In particular, we show that the mapping has a fixed point if and only if the problem has a solution. Then, using this mapping, we propose a splitting method for solving the problem in a real Hilbert space. In particular, we show that the sequences generated by the method converge strongly to the solution to the problem under certain assumptions. Convergence rate analysis of the methods is also provided to illustrate the method's efficiency. Finally, we apply the results to a class of optimization problems.