A new strongly convergent algorithm to solve pseudo-monotone equilibrium problems in a real Hilbert space

We propose a new strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of common fixed points of a family of strict pseudocontraction mappings in a real Hilbert space. The strong convergence theorem of proposed algorithms is investigated without the Lipschitz condition for the bifunctions. Our results complement many known recent results in the literature.

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

Symmetry ◽

10.3390/sym11040480 ◽

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.

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On Computability and Applicability of Mann-Reich-Sabach-Type Algorithms for Approximating the Solutions of Equilibrium Problems in Hilbert Spaces

Abstract and Applied Analysis ◽

10.1155/2018/7218487 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9

Author(s):

F. O. Isiogugu ◽

P. Pillay ◽

P. U. Nwokoro

Keyword(s):

Hilbert Space ◽

Hilbert Spaces ◽

Equilibrium Problems ◽

Real Hilbert Space ◽

Contemporary Literature ◽

Iteration Scheme ◽

Common Elements ◽

Type Mapping ◽

The Common ◽

Selection Of

We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points F(T) of a multivalued (or single-valued) k-strictly pseudocontractive-type mapping T and the set of solutions EP(F) of an equilibrium problem for a bifunction F in a real Hilbert space H. This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence {Kn}n=1∞ of closed convex subsets of H from an arbitrary x0∈H and a sequence {xn}n=1∞ of the metric projections of x0 into Kn. The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.

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A New Iterative Method for Equilibrium Problems and Fixed Point Problems

Abstract and Applied Analysis ◽

10.1155/2013/178053 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Abdul Latif ◽

Mohammad Eslamian

Keyword(s):

Hilbert Space ◽

Fixed Point ◽

Variational Inequality ◽

Iterative Method ◽

Equilibrium Problems ◽

Variational Inequality Problem ◽

Real Hilbert Space ◽

Common Element ◽

Continuous Mappings ◽

New Iterative Method

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

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An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications

Axioms ◽

10.3390/axioms9030099 ◽

2020 ◽

Vol 9 (3) ◽

pp. 99 ◽

Cited By ~ 2

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.

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Strong Convergence Theorems for Equilibrium Problems andk-Strict Pseudocontractions in Hilbert Spaces

Abstract and Applied Analysis ◽

10.1155/2011/276874 ◽

2011 ◽

Vol 2011 ◽

pp. 1-13 ◽

Cited By ~ 3

Author(s):

Dao-Jun Wen

Keyword(s):

Hilbert Space ◽

Fixed Point ◽

Strong Convergence ◽

Hilbert Spaces ◽

Equilibrium Problems ◽

Iterative Scheme ◽

Real Hilbert Space ◽

Finite Family ◽

Common Element ◽

Convergence Theorems

We introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of a finite family ofk-strictly pseudo-contractive nonself-mappings. Strong convergence theorems are established in a real Hilbert space under some suitable conditions. Our theorems presented in this paper improve and extend the corresponding results announced by many others.

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An inertial extragradient method for solving bilevel equilibrium problems

Carpathian Journal of Mathematics ◽

10.37193/cjm.2020.01.09 ◽

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.

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

Demonstratio Mathematica ◽

10.1515/dema-2021-0030 ◽

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.

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Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications

AIMS Mathematics ◽

10.3934/math.2021093 ◽

2021 ◽

Vol 6 (2) ◽

pp. 1538-1560

Author(s):

Habib ur Rehman ◽

◽

Poom Kumam ◽

Kanokwan Sitthithakerngkiet ◽

◽

...

Keyword(s):

Hilbert Space ◽

Equilibrium Problems ◽

Real Hilbert Space ◽

Type Method

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Inertial Iterative Self-Adaptive Step Size Extragradient-Like Method for Solving Equilibrium Problems in Real Hilbert Space with Applications

Axioms ◽

10.3390/axioms9040127 ◽

2020 ◽

Vol 9 (4) ◽

pp. 127

Author(s):

Wiyada Kumam ◽

Kanikar Muangchoo

Keyword(s):

Hilbert Space ◽

Variational Inequality ◽

Equilibrium Problems ◽

Numerical Study ◽

Real Hilbert Space ◽

Convergence Result ◽

Variational Inequality Problems ◽

Test Problems ◽

Step Size ◽

The Cost

A number of applications from mathematical programmings, such as minimization problems, variational inequality problems and fixed point problems, can be written as equilibrium problems. Most of the schemes being used to solve this problem involve iterative methods, and for that reason, in this paper, we introduce a modified iterative method to solve equilibrium problems in real Hilbert space. This method can be seen as a modification of the paper titled “A new two-step proximal algorithm of solving the problem of equilibrium programming” by Lyashko et al. (Optimization and its applications in control and data sciences, Springer book pp. 315–325, 2016). A weak convergence result has been proven by considering the mild conditions on the cost bifunction. We have given the application of our results to solve variational inequality problems. A detailed numerical study on the Nash–Cournot electricity equilibrium model and other test problems is considered to verify the convergence result and its performance.

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