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

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.