The forward-backward-forward algorithm with extrapolation from the past and penalty scheme for solving monotone inclusion problems and applications

TL;DR

This paper introduces an enhanced iterative algorithm for solving complex monotone inclusion problems, extending previous methods with extrapolation and penalty schemes, and demonstrates its effectiveness through theoretical analysis and a practical image inpainting application.

Contribution

The paper develops a novel forward-backward-forward algorithm with extrapolation and penalty schemes, extending existing methods and applying it to various optimization and inclusion problems.

Findings

Proven weak ergodic convergence of the algorithm.

Established strong convergence when the operator is strongly monotone.

Validated the method with a TV-based image inpainting numerical experiment.

Abstract

In this paper, we propose an improved iterative method for solving the monotone inclusion problem in the form of $0 \in Ax + Dx + N_{C}(x)$ in real Hilbert space, where $A$ is a maximally monotone operator, $D$ and $B$ are monotone and Lipschitz continuous, and $C$ is the nonempty set of zeros of the operator $B$. Our investigated method, called Tseng's forward-backward-forward with extrapolation from the past and penalty scheme, extends the one proposed by Bot and Csetnek [Set-Valued Var. Anal. 22: 313--331, 2014]. We investigate the weak ergodic and strong convergence (when $A$ is strongly monotone) of the iterates produced by our proposed scheme. We show that the algorithmic scheme can also be applied to minimax problems. Furthermore, we discuss how to apply the method to the inclusion problem involving a finite sum of compositions of linear continuous operators by using the product space approach and employ it for convex minimization. Finally, we present a numerical experiment in TV-based image inpainting to validate the proposed theoretical theorem.