scholarly journals Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 822 ◽  
Author(s):  
Habib ur Rehman ◽  
Poom Kumam ◽  
Ioannis K. Argyros ◽  
Meshal Shutaywi ◽  
Zahir Shah
Keyword(s):  
Variational Inequality ◽  
Practical Reason ◽  
Line Search ◽  
Equilibrium Problems ◽  
Convergence Result ◽  
Proximal Methods ◽  
Search Procedure ◽  
Equilibrium Models ◽  
Convergence Results ◽  
Weak Convergence Theorem

In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods.

scholarly journals Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models

Energies ◽  
2020 ◽  
Vol 13 (12) ◽  
pp. 3292 ◽  
Author(s):  
Habib ur Rehman ◽  
Poom Kumam ◽  
Meshal Shutaywi ◽  
Nasser Aedh Alreshidi ◽  
Wiyada Kumam
Keyword(s):  
Variational Inequality ◽  
Equilibrium Problems ◽  
Extragradient Method ◽  
Mathematical Finance ◽  
Convergence Result ◽  
Variational Inequality Problems ◽  
Subgradient Extragradient Method ◽  
Pseudomonotone Operator ◽  
Equilibrium Models ◽  
Variable Stepsize

This manuscript aims to incorporate an inertial scheme with Popov’s subgradient extragradient method to solve equilibrium problems that involve two different classes of bifunction. The novelty of our paper is that methods can also be used to solve problems in many fields, such as economics, mathematical finance, image reconstruction, transport, elasticity, networking, and optimization. We have established a weak convergence result based on the assumption of the pseudomonotone property and a certain Lipschitz-type cost bifunctional condition. The stepsize, in this case, depends upon on the Lipschitz-type constants and the extrapolation factor. The bifunction is strongly pseudomonotone in the second method, but stepsize does not depend on the strongly pseudomonotone and Lipschitz-type constants. In contrast, the first convergence result, we set up strong convergence with the use of a variable stepsize sequence, which is decreasing and non-summable. As the application, the variational inequality problems that involve pseudomonotone and strongly pseudomonotone operator are considered. Finally, two well-known Nash–Cournot equilibrium models for the numerical experiment are reviewed to examine our convergence results and show the competitive advantage of our suggested methods.

scholarly journals Weak convergence of explicit extragradient algorithms for solving equilibirum problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Habib ur Rehman ◽  
Poom Kumam ◽  
Yeol Je Cho ◽  
Pasakorn Yordsorn
Keyword(s):  
Weak Convergence ◽  
Line Search ◽  
Equilibrium Problems ◽  
Search Procedure ◽  
Step Size ◽  
Convergence Results ◽  
Size Evaluation ◽  
New Algorithms ◽  
Number Of Iterations ◽  
Over Time

Abstract This paper aims to propose two new algorithms that are developed by implementing inertial and subgradient techniques to solve the problem of pseudomonotone equilibrium problems. The weak convergence of these algorithms is well established based on standard assumptions of a cost bi-function. The advantage of these algorithms was that they did not need a line search procedure or any information on Lipschitz-type bifunction constants for step-size evaluation. A practical explanation for this is that they use a sequence of step-sizes that are updated at each iteration based on some previous iterations. For numerical examples, we discuss two well-known equilibrium models that assist our well-established convergence results, and we see that the suggested algorithm has a competitive advantage over time of execution and the number of iterations.

scholarly journals PARALLEL EXTRAGRADIENT-PROXIMAL METHODS FOR SPLIT EQUILIBRIUM PROBLEMS

2016 ◽  
Vol 21 (4) ◽  
pp. 478-501 ◽  
Author(s):  
Dang Van Hieu
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Projection Method ◽  
Equilibrium Problems ◽  
Extragradient Method ◽  
Variational Inequality Problems ◽  
Proximal Methods ◽  
Proximal Method ◽  
Weak And Strong Convergence ◽  
Split Variational Inequality

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.

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

Axioms ◽  
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.

scholarly journals The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 463 ◽  
Author(s):  
Habib ur Rehman ◽  
Poom Kumam ◽  
Wiyada Kumam ◽  
Meshal Shutaywi ◽  
Wachirapong Jirakitpuwapat
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Gradient Method ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Considerable Influence ◽  
Numerical Examples ◽  
Convergence Results ◽  
Monotone Variational Inequality ◽  
Number Of Iterations

In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We also present the application of our results that enable to solve numerically the pseudo-monotone and monotone variational inequality problems, in addition to the particular presumptions required by the operator. We have used various numerical examples to support our well-proved convergence results, and we can show that the proposed method involves a considerable influence over-running time and the total number of iterations.

scholarly journals Solving common nonmonotone equilibrium problems using an inertial parallel hybrid algorithm with Armijo line search with applications to image recovery

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Suthep Suantai ◽  
Damrongsak Yambangwai ◽  
Watcharaporn Cholamjiak
Keyword(s):  
Hybrid Algorithm ◽  
Line Search ◽  
Equilibrium Problems ◽  
Common Solution ◽  
Parallel Hybrid ◽  
Armijo Line Search ◽  
Weak Convergence Theorem ◽  
The Common ◽  
Convexity Assumptions

AbstractIn this work, we modify the inertial hybrid algorithm with Armijo line search using a parallel method to approximate a common solution of nonmonotone equilibrium problems in Hilbert spaces. A weak convergence theorem is proved under some continuity and convexity assumptions on the bifunction and the nonemptiness of the common solution set of Minty equilibrium problems. Furthermore, we demonstrate the quality of our inertial parallel hybrid algorithm by using image restoration, as well as its superior efficiency when compared with previously considered parallel algorithms.

scholarly journals Construction of a common element for the set of solutions of fixed point problems and generalized equilibrium problems in Hilbert spaces

2016 ◽  
Vol 15 (1) ◽  
pp. 79-96
Author(s):  
Muhammad Aqeel Ahmad Khan
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Finite Family ◽  
Common Element ◽  
Common Solution ◽  
Convergence Results ◽  
Generalized Equilibrium

AbstractIn this paper, we propose and analyse an iterative algorithm for the approximation of a common solution for a finite family of k-strict pseudocontractions and two finite families of generalized equilibrium problems in the setting of Hilbert spaces. Strong convergence results of the proposed iterative algorithm together with some applications to solve the variational inequality problems are established in such setting. Our results generalize and improve various existing results in the current literature.

scholarly journals Inertial method for split null point problems with pseudomonotone variational inequality problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chibueze Christian Okeke ◽  
Abdulmalik Usman Bello ◽  
Lateef Olakunle Jolaoso ◽  
Kingsley Chimuanya Ukandu
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Convergence Result ◽  
Null Point ◽  
Point Problem ◽  
Inertial Method ◽  
Convergence Results ◽  
Split Null Point Problem ◽  
Self Adaptive

<p style='text-indent:20px;'>This paper analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space. Furthermore, in this study, a strong convergence result is obtained without assuming Lipschitz continuity of the associated mapping and the operator norm is self adaptive. Additionally, the proposed algorithm only uses one projections onto the feasible set in each iteration. More so, the strong convergence results are obtained under some relaxed conditions on the initial factor and the iterative parameters. Numerical results are presented to illustrate the performance of the proposed algorithm.The results obtained in this study improved and extended related studies in the literature.</p>

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 A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

2021 ◽  
Author(s):  
Stefano Almi ◽  
Marco Morandotti ◽  
Francesco Solombrino
Keyword(s):  
Mean Field ◽  
Evolutionary Games ◽  
Type Systems ◽  
Convergence Result ◽  
Performance Criterion ◽  
Mean Field Limit ◽  
Spatially Inhomogeneous ◽  
Convergence Results ◽  
Special Cases ◽  
Lagrangian Scheme

AbstractA multi-step Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics, and is a multi-step scheme, as it develops on the following two stages: First, the distribution of strategies or labels is updated according to a best performance criterion, and then, this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.

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