A new preconditioning algorithm for finding a zero of the sum of two monotone operators and its application to image restoration problem

TL;DR

This paper introduces a novel preconditioning forward-backward algorithm for finding zeros of sums of monotone operators, with proven strong convergence and applications to convex minimization and image restoration.

Contribution

It presents a new algorithm that broadens existing methods, specifically handling one maximal monotone and one M-cocoercive operator, with proven convergence in Hilbert spaces.

Findings

The algorithm converges strongly in Hilbert spaces.

It outperforms existing algorithms in convex minimization tasks.

Applications to image restoration demonstrate practical effectiveness.

Abstract

Finding a zero of the sum of two monotone operators is one of the most important problems in monotone operator theory, and the forward-backward algorithm is the most prominent approach for solving this type of problem. The aim of this paper is to present a new preconditioning forward-backward algorithm to obtain the zero of the sum of two operators in which one is maximal monoton and the other one is M-cocoercive, where M is a linear bounded operator. Furthermore, the strong convergence of the proposed algorithm, which is a broader variant of previously known algorithms, has been proven in Hilbert spaces. We also use our algorithm to tackle the convex minimization problem and show that it outperforms existing algorithms. Finally, we discuss several image restoration applications.