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

ABSTRACT

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.

Keywords:

• Splitting method
• maximal monotone
• resolvent
• fixed point
• nonexpansive
• Hilbert space

2010 Mathematics Subject Classifications:

• 47H05
• 47H09
• 47H10
• 47J25
• 90C25

Acknowledgments

The author is grateful to Professors W. Takahashi of Tokyo Institute of Technology, D. Kuroiwa of Shimane university and Li Xu of Akita Prefectural University for their helpful support. We thank the Associate Editor and the reviewers for their very helpful comments. In particular, we would like to thank one of the reviewers for the comments that help improve the paper, especially for pointing out the relation between the proposed method and the straightforward application of the Douglas–Rachford method in Remark 5.4. This work was supported in part by the Ministry of Education, Culture, Sports, Science, and Technology (grant numbers 16K05280 and 19K03639).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the Ministry of Education, Culture, Sports, Science, and Technology [grant numbers 16K05280,19K03639].