Relaxed Inertial Tseng’s Type Method for Solving the Inclusion Problem with Application to Image Restoration

Abstract

The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the inertial extrapolation step. The proposed method follows from time explicit discretization of a dynamical system. A weak convergence of the iterates generated by the method involving monotone operators is given. Moreover, the iterative scheme uses a variable step size, which does not depend on the Lipschitz constant of the underlying operator given by a simple updating rule. Furthermore, the proposed algorithm is modified and used to derive a scheme for solving a split feasibility problem. The proposed schemes are used in solving the image deblurring problem to illustrate the applicability of the proposed methods in comparison with the existing state-of-the-art methods.

Projection method with inertial step for nonlinear equations: Application to signal recovery

“…Polyak [54] introduced an inertial extrapolation as an acceleration mechanism to solve the smooth convex minimization problem based on the heavy ball methods of the two-order time dynamical system. The inertial algorithm is a two-step iterative method that uses the previous two iterates to compute the next iterate, and it can be thought of as a procedure of speeding up the convergence properties, see [54,17,14,5,15]. An abundant literature has also been devoted on this subject with a lot of fast iterative algorithms constructed using the inertial extrapolation, for instance, inertial forward-backward splitting methods [18,50], inertial Mann method [58], inertial Douglas-Rachford splitting method [24], inertial ADMM [25], inertial subgradient extragradient method [59], inertial forward-backward-forward method [22], inertial proximal-extragradient method [23], and inertial contraction method [32].…”

A Modified Tseng’s Method for Solving the Modified Variational Inclusion Problems and Its Applications

“…Attouch and Cabot [21,22] established relaxed inertial proximal algorithms by using both techniques to increase the performance of algorithms to solve the previous problems. Later, many researchers focused on both techniques and introduced the relaxed inertial forward-backward algorithm [23], the inertial Douglas-Rachford algorithm [24], a Tseng extragradient method in Banach spaces [25], and the relaxed inertial Tseng's type algorithm [26]. Among the well-known algorithms is the relaxed inertial Tseng's type method developed by Abubakar et al [26].…”

An inertial iterative scheme for solving variational inclusion with application to Nash-Cournot equilibrium and image restoration problems

“…Recently, the authors in [41] successfully introduced another modified forward backward method called forward-reflected-backward splitting method which also does not requires the cocoercivity assumption on the operator B. Although, the convergence of the modified forward-backward schemes in [58] and [41] are formulated without the strong assumptions, it can be observed that the stepsizes considered in these works are either a fixed stepsize chosen in (0, 1 L ) (L is the Lipschitz constant of B) or the stepsize can be computed using a line search procedure with finite stopping criterion. It is known that, line search procedures involve extra functions evaluations or computations of resolvent of the considered operator, thereby reducing the computational performance of a given scheme.…”

“…It is known that, line search procedures involve extra functions evaluations or computations of resolvent of the considered operator, thereby reducing the computational performance of a given scheme. A modification of the forward-reflected-backward splitting method is proposed in [61] by considering variable stepsizes which are updated over each iteration by some simple computations given as: choose λ 0 > 0 and ν ∈ (0, 1 2 ) such that…”

“…Several variants of the forward-backward-forward method have been studied and a number of schemes have been presented to solve (1.1) (Refs. [1,6,19,23,24,27,32,33,49,52,53,55,58]). Inertial extrapolation techniques are introduced as a process of accelerating the convergence rate of an iterative scheme.…”