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

Abstract

The goal of this study was to show how a modified variational inclusion problem can be solved based on Tseng’s method. In this study, we propose a modified Tseng’s method and increase the reliability of the proposed method. This method is to modify the relaxed inertial Tseng’s method by using certain conditions and the parallel technique. We also prove a weak convergence theorem under appropriate assumptions and some symmetry properties and then provide numerical experiments to demonstrate the convergence behavior of the proposed method. Moreover, the proposed method is used for image restoration technology, which takes a corrupt/noisy image and estimates the clean, original image. Finally, we show the signal-to-noise ratio (SNR) to guarantee image quality.

An Inertial Forward–Backward Splitting Method for Solving Modified Variational Inclusion Problems and Its Application

“…where operators S : H → H and T : H → 2 H . Tseng's technique [1,2], the proximal point method [3][4][5], the forward-backward splitting method [6][7][8][9] and other methods for the variational inclusion problem have received great attention from an increasing number of researchers. The forward-backward splitting method is one of the most commonly used.…”

“…Secondly, the convergence theorem was demonstrated for non-smooth convex minimization problems and monotone inclusions. The relaxed inertial forward-backward methods [10,11], the inertial proximal point algorithm [12,13] and the inertial Tseng's type method [2,14] were all created as a consequence of this concentration on both methods.…”

A Regularized Tseng Method for Solving Various Variational Inclusion Problems and Its Application to a Statistical Learning Model

“…They proposed a Halperninertial forward-backward splitting algorithm for solving the problem and proved a strong convergence theorem. Moreover, the authors of [18] studied the MVIP in the case where the T i s are monotone and Lipschitz continuous and designed a modified Tseng method for solving it. By using some symmetry properties, they proved the weak convergence of the method they proposed.…”

Tseng‐Type Algorithms for the Split Variational Inclusion

“…Variational inclusion is at the core of the modeling of many problems, such as variational inequalities [2][3][4], optimization problems [5], split problems [6][7][8][9], equilibrium problems [10], and xed point problems [11]. Variational inclusion (1) has been extended and studied in di erent ways, see [12][13][14][15][16][17][18]. An e cient way for solving (1) is the forward-backward iterate [19][20][21] de ned by…”

“…Note that the Lipschitz constant of f is usually unknown or di cult to estimate in many problems. Tseng's method has been applied extensively, and some self-adaptive techniques are used, see [14,[23][24][25][26]. Especially, Cholamjiak, Hieu, and Cho [23] suggested an iterative method described as follows:…”