Weak convergence of explicit extragradient algorithms for solving equilibirum problems

• Published in: Journal of inequalities and applications Vol. 2019; no. 1; pp. 1 - 25
• Main Authors: ur Rehman, Habib, Kumam, Poom, Cho, Yeol Je, Yordsorn, Pasakorn
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 07.11.2019; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Analysis; Applications of Mathematics; Convergence; Economic models; Equilibrium problem; Extragradient method; Iterative methods; Lipschitz-type conditions; Mathematics; Mathematics and Statistics; Nash–Cournot equilibrium model of electricity markets; 65K15; 47H10; Equilibrium problem; Lipschitz-type conditions; Extragradient method; 47H05; 65Y05; Nash–Cournot equilibrium model of electricity markets
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-019-2233-1

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.