# An inertial three-operator splitting algorithm with applications to image inpainting

## TL;DR

This paper introduces an inertial three-operator splitting algorithm with proven convergence, applied to convex minimization problems and demonstrated to be effective in image inpainting with nuclear norm regularization.

## Contribution

It proposes a novel inertial splitting algorithm with convergence analysis and applies it to image inpainting, showing improved performance over existing methods.

## Key findings

- Convergence of the inertial algorithm is established under specific conditions.
- Numerical experiments show the algorithm's effectiveness in image inpainting.
- The method outperforms some existing algorithms in terms of efficiency.

## Abstract

The three-operators splitting algorithm is a popular operator splitting method for finding the zeros of the sum of three maximally monotone operators, with one of which is cocoercive operator. In this paper, we propose a class of inertial three-operator splitting algorithm. The convergence of the proposed algorithm is proved by applying the inertial Krasnoselskii-Mann iteration under certain conditions on the iterative parameters in real Hilbert spaces. As applications, we develop an inertial three-operator splitting algorithm to solve the convex minimization problem of the sum of three convex functions, where one of them is differentiable with Lipschitz continuous gradient. Finally, we conduct numerical experiments on a constrained image inpainting problem with nuclear norm regularization. Numerical results demonstrate the advantage of the proposed inertial three-operator splitting algorithms.