Weak and strong convergence of prox-penalization and splitting algorithms  for   bilevel equilibrium problems

In this paper, we introduce two parallel extragradient-proximal methods for solving split equilibrium problems. The algorithms combine the extragradient method, the proximal method and the shrinking projection method. The weak and strong convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for equilibrium bifunctions. We also present an application to split variational inequality problems and a numerical example to illustrate the convergence of the proposed algorithms.

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Inertial Extragradient Methods for Solving Split Equilibrium Problems

Mathematics ◽

10.3390/math9161884 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1884

Author(s):

Suthep Suantai ◽

Narin Petrot ◽

Manatchanok Khonchaliew

Keyword(s):

Strong Convergence ◽

Hilbert Spaces ◽

Numerical Experiments ◽

Equilibrium Problems ◽

Constraint Qualifications ◽

Weak And Strong Convergence ◽

Convergence Theorems ◽

Extragradient Methods

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms.

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Modified inertial subgradient extragradient method for equilibrium problems

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2021-0099 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Lateef Olakunle Jolaoso ◽

Yekini Shehu ◽

Regina N. Nwokoye

Keyword(s):

Variational Inequalities ◽

Strong Convergence ◽

Equilibrium Problems ◽

Extragradient Method ◽

Subgradient Extragradient Method ◽

Weak And Strong Convergence ◽

Extragradient Methods ◽

Convergence Results ◽

Inertial Extrapolation ◽

Special Case

Abstract The subgradient extragradient method with inertial extrapolation step x n + θ n (x n − x n−1) (also known as inertial subgradient extragradient method) has been studied extensively in the literature for solving variational inequalities and equilibrium problems. Most of the inertial subgradient extragradient methods in the literature for both variational inequalities and equilibrium problems have not considered the special case when the inertial factor θ n = 1. The convergence results have always been obtained when the inertial factor θ n is assumed 0 ≤ θ n < 1. This paper considers the relaxed inertial version of subgradient extragradient method for equilibrium problems with 0 ≤ θ n ≤ 1. We give both weak and strong convergence results using this inertial subgradient extragradient method and also give some numerical illustrations.

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Weak and Strong Convergence Theorems for Equilibrium Problems and Countable Strict Pseudocontractions Mappings in Hilbert Space

Journal of Inequalities and Applications ◽

10.1155/2010/474813 ◽

2010 ◽

Vol 2010 ◽

pp. 1-11 ◽

Cited By ~ 5

Author(s):

Rudong Chen ◽

Xilin Shen ◽

Shujun Cui

Keyword(s):

Hilbert Space ◽

Strong Convergence ◽

Equilibrium Problems ◽

Weak And Strong Convergence ◽

Convergence Theorems

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Iterative Methods of Weak and Strong Convergence Theorems for the Split Common Solution of the Feasibility Problems, Generalized Equilibrium Problems, and Fixed Point Problems

Journal of Mathematics ◽

10.1155/2020/6689015 ◽

2020 ◽

Vol 2020 ◽

pp. 1-22

Author(s):

Shahram Rezapour ◽

Yuanheng Wang ◽

Seyyed Hasan Zakeri

Keyword(s):

Fixed Point ◽

Strong Convergence ◽

Equilibrium Problems ◽

Nonexpansive Mappings ◽

Equilibrium Points ◽

Fixed Point Problems ◽

Weak And Strong Convergence ◽

Iterative Approximation ◽

Common Solution ◽

Generalized Equilibrium

The purpose of this paper is to introduce the extragradient methods for solving split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings. We establish the results of weak and strong convergence under appropriate conditions. As applications of our three main theorems, when the mappings and their domains take different types of cases, we can obtain nine iterative approximation theorems and corollas on fixed points, variational inequality solutions, and equilibrium points.

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