Bounded perturbation resilience of a regularized forward-reflected-backward splitting method for solving variational inclusion problems with applications

ABSTRACT

The forward-reflected-backward splitting method recently introduced for solving variational inclusion problems involves just one forward evaluation and one backward evaluation of the monotone operator and the maximal monotone operator, respectively, per iteration. This structure gives it some advantage over the earlier proposed methods. However, it only provides weak convergence, in general. Our aim in this paper is to improve the forward-reflected-backward splitting method in order to obtain strong convergence. To this end, we first study a regularized variational inclusion problem of finding the zero of the sum of two monotone operators. We then propose a regularized forward-reflected-backward splitting method for approximating a solution to the problem and prove the strong convergence of our iterative scheme under some suitable assumptions on the parameters. Moreover, we show that our algorithm has the bounded perturbation resilience property. Furthermore, we apply our results to convex minimization, split feasibility, split variational inclusion, and image deblurring problems, and illustrate the performance of our algorithm with several numerical examples.

KEYWORDS:

• Bounded perturbation resilience
• monotone operator
• regularization
• strong convergence

Acknowledgments

Both authors are grateful to the editors and the referees for their useful comments and helpful suggestions.

Data availability

All the data in this paper which were generated from the MATLAB codes are available from the corresponding author upon request.

Disclosure statement

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

Additional information

Funding

Simeon Reich was partially supported by the Israel Science Foundation [grant number 820/17], by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.