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 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 Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 881 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Xiaolong Qin ◽  
Yekini Shehu ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Subgradient Extragradient Method ◽  
Pseudomonotone Operator ◽  
Common Solution ◽  
Asymptotically Nonexpansive

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI).

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.

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