Strong Convergence of Forward-Reflected-Backward Splitting Methods for Solving Monotone Inclusions with Applications to Image Restoration and Optimal Control

In this paper, we propose and study several strongly convergent versions of the forward-reflected-backward splitting method of Malitsky and Tam for finding a zero of the sum of two monotone operators in a real Hilbert space. Our proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. We also develop inertial versions of our methods and strong convergence results are obtained for these methods when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous and monotone. Finally, we discuss some examples from image restorations and optimal control regarding the implementations of our methods in comparisons with known related methods in the literature.

Automated summary

In this paper, several strongly convergent versions of the forward-reflected-backward splitting method are proposed and studied for finding a zero of the sum of two monotone operators in a real Hilbert space. The proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration, which is a feature absent in many other available strongly convergent splitting methods. Inertial versions of the methods are also developed, and strong convergence results are obtained for these methods under certain conditions on the operators. Examples from image restorations and optimal control are discussed to demonstrate the effectiveness of the proposed methods compared to existing methods in the literature.