A forward-reflected-backward algorithm with momentum terms for solving bilevel monotone inclusion problems with an application

Abstract

This paper introduces a regularized forward-reflected-backward splitting method with momentum terms for solving bilevel monotone inclusion problems (BMIP). The proposed method is a modification and extension of the forward-reflected-backward splitting method studied by Malitsky and Tam for solving monotone inclusion problems, and it inherits an interesting feature of one forward evaluation and one backward evaluation of the monotone operator and maximal monotone operator, respectively, per iteration. In contrast to this method, the iterative sequences generated by our method converge strongly to a solution of the problem studied in this work. Additionally, the momentum terms incorporated in our method improve its convergence behaviour. Finally, we apply our method to image restoration and demonstrate the applicability and performance of our method through numerical experiments.