scholarly journals A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1165 ◽  
Author(s):  
Pasakorn Yordsorn ◽  
Poom Kumam ◽  
Habib ur Rehman ◽  
Abdulkarim Hassan Ibrahim
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Line Search ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Adaptive Method ◽  
Type Condition ◽  
Computational Performance ◽  
Weak Convergence Theorem

In this paper, we presented a modification of the extragradient method to solve pseudomonotone equilibrium problems involving the Lipschitz-type condition in a real Hilbert space. The method uses an inertial effect and a formula for stepsize evaluation, that is updated for each iteration based on some previous iterations. The key advantage of the algorithm is that it is achieved without previous knowledge of the Lipschitz-type constants and also without any line search procedure. A weak convergence theorem for the proposed method is well established by assuming mild cost bifunction conditions. Many numerical experiments are presented to explain the computational performance of the method and to equate them with others.

scholarly journals An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 99 ◽  
Author(s):  
Nopparat Wairojjana ◽  
Habib ur Rehman ◽  
Ioannis K. Argyros ◽  
Nuttapol Pakkaranang
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Numerical Solution ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Suggested Method ◽  
Experimental Results ◽  
Weak Convergence Theorem

Several methods have been put forward to solve equilibrium problems, in which the two-step extragradient method is very useful and significant. In this article, we propose a new extragradient-like method to evaluate the numerical solution of the pseudomonotone equilibrium in real Hilbert space. This method uses a non-monotonically stepsize technique based on local bifunction values and Lipschitz-type constants. Furthermore, we establish the weak convergence theorem for the suggested method and provide the applications of our results. Finally, several experimental results are reported to see the performance of the proposed method.

scholarly journals Convergence Analysis of Self-Adaptive Inertial Extra-Gradient Method for Solving a Family of Pseudomonotone Equilibrium Problems with Application

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1332
Author(s):  
Thanatporn Bantaojai ◽  
Nuttapol Pakkaranang ◽  
Habib ur Rehman ◽  
Poom Kumam ◽  
Wiyada Kumam
Keyword(s):  
Hilbert Space ◽  
Line Search ◽  
Fixed Point Theorems ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Search Method ◽  
Type Condition ◽  
Variable Stepsize ◽  
Mild Conditions ◽  
Line Search Method

In this article, we propose a new modified extragradient-like method to solve pseudomonotone equilibrium problems in real Hilbert space with a Lipschitz-type condition on a bifunction. This method uses a variable stepsize formula that is updated at each iteration based on the previous iterations. The advantage of the method is that it operates without prior knowledge of Lipschitz-type constants and any line search method. The weak convergence of the method is established by taking mild conditions on a bifunction. In the context of an application, fixed-point theorems involving strict pseudo-contraction and results for pseudomonotone variational inequalities are considered. Many numerical results have been reported to explain the numerical behavior of the proposed method.

scholarly journals Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 480
Author(s):  
Manatchanok Khonchaliew ◽  
Ali Farajzadeh ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Numerical Experiments ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Constraint Qualifications ◽  
Solution Sets ◽  
New Algorithms

This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, these two new algorithms show strong convergence. Some numerical experiments are presented to demonstrate the new algorithms. Finally, the two introduced algorithms are compared with a standard, well-known algorithm.

scholarly journals An inertial extragradient method for solving bilevel equilibrium problems

2020 ◽  
Vol 36 (1) ◽  
pp. 91-107
Author(s):  
JIRAPRAPA MUNKONG ◽  
BUI VAN DINH ◽  
KASAMSUK UNGCHITTRAKOOL
Keyword(s):  
Hilbert Space ◽  
Numerical Experiment ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Iteration Process ◽  
Iterative Sequence ◽  
Inertial Term ◽  
Speed Up ◽  
Bilevel Equilibrium Problems

In this paper, we propose an algorithm with two inertial term extrapolation steps for solving bilevel equilibrium problem in a real Hilbert space. The inertial term extrapolation step is introduced to speed up the rate of convergence of the iteration process. Under some sufficient assumptions on the bifunctions involving pseudomonotone and Lipschitz-type conditions, we obtain the strong convergence of the iterative sequence generated by the proposed algorithm. A numerical experiment is performed to illustrate the numerical behavior of the algorithm and also comparison with some other related algorithms in the literature.

scholarly journals Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

2021 ◽  
Vol 54 (1) ◽  
pp. 280-298
Author(s):  
Nuttapol Pakkaranang ◽  
Habib ur Rehman ◽  
Wiyada Kumam
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Type Condition ◽  
Step Size ◽  
Extragradient Methods ◽  
Iterative Schemes ◽  
Mild Conditions

Abstract The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

scholarly journals Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 76
Author(s):  
Chainarong Khanpanuk ◽  
Nuttapol Pakkaranang ◽  
Nopparat Wairojjana ◽  
Nattawut Pholasa
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Inertial Term ◽  
Mild Conditions ◽  
Lipschitz Constants

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature.

scholarly journals WEAK CONVERGENCE OF AN ITERATIVE SCHEME FOR GENERALIZED EQUILIBRIUM PROBLEMS

2009 ◽  
Vol 79 (3) ◽  
pp. 437-453 ◽  
Author(s):  
JIAN-WEN PENG ◽  
JEN-CHIH YAO
Keyword(s):  
Weak Convergence ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Extragradient Method ◽  
Continuous Mapping ◽  
Common Element ◽  
Lipschitz Continuous ◽  
Lipschitz Continuous Mapping ◽  
Generalized Equilibrium ◽  
Weak Convergence Theorem

AbstractIn this paper, we introduce an iterative scheme using an extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz-continuous mapping. We obtain a weak convergence theorem for three sequences generated by this process. Based on this result, we also obtain several interesting results. The results in this paper generalize and extend some well-known weak convergence theorems in the literature.

scholarly journals Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems

2022 ◽  
Vol 12 (1) ◽  
pp. 63
Author(s):  
Do Sang Kim ◽  
Nguyen Ngoc Hai ◽  
Bui Van Dinh
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Proximal Point Algorithm ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Common Point ◽  
Hybrid Mapping ◽  
Proximal Point ◽  
Extragradient Algorithm ◽  
Generalized Hybrid Mapping

<p style='text-indent:20px;'>In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.</p>

scholarly journals A Mean Extragradient Method for Solving Variational Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 462
Author(s):  
Apichit Buakird ◽  
Nimit Nimana ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Numerical Experiments ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Mean Value ◽  
Inequality Problem ◽  
The Mean ◽  
Mean Value Method

We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann’s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.

Sign in / Sign up

Export Citation Format

Share Document