2020
DOI: 10.3390/math8050818
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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… Show more

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Cited by 23 publications
(23 citation statements)
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“…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].…”
Section: Introductionmentioning
confidence: 99%
“…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].…”
Section: Introductionmentioning
confidence: 99%
“…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].…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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…”
Section: Introductionmentioning
confidence: 99%
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