A new Popov's subgradient extragradient method for two classes of equilibrium programming in a real Hilbert space

Abstract

In this paper, we proposed two different methods for solving pseudomonotone and strongly pseudomonotone equilibrium problems. We can examine these methods as an extension and improvement of the Popov's extragradient method. We replaced the second minimization problem onto a closed convex set in the Popov's extragradient method, with a half-space minimization problem that is updated on each iteration and also formulates a useful method for determining the appropriate stepsize on each iteration. The weak convergence theorem of the first method and strong convergence theorem for the second method is well-established based on a standard assumption on a cost bifunction. We also consider various numerical examples to support our well-established convergence results, and we can see that the proposed methods depict a significant improvement in terms of the number of iterations and execution time.

Keywords:

• Equilibrium problem
• pseudomonotone bifunction
• strongly pseudomonotone bifunction
• Lipschitz-type conditions
• variational inequality problems

2010 Mathematics Subject Classifications:

• 47H06
• 47H09
• 47J05
• 47J25

Acknowledgements

The first author would like to thank the ‘Petchra Pra Jom Klao Ph.D. Research Scholarship’ from King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand. We are very grateful to assistant editor and the anonymous referees for their valuable and useful comments, which helps in improving the quality of this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Two matrices are randomly generated E and F with entries from $[-1,1].$ The matrix $B=E^{T}E$, $S=F^{T}F$ and $A=S+B.$

Additional information

Funding

This research work was financially supported by King Mongkut’s University of Technology Thonburi through the ‘KMUTT 55thAnniversary Commemorative Fund’. Moreover, this project was supported by Theoretical and Computational Science (TaCS) Center underComputational and Applied Science for Smart research Innovation research Cluster (CLASSIC), Faculty of Science, KMUTT. In particular, Habib ur Rehman was financed by the Petchra Pra Jom Doctoral Scholarship Academic for Ph.D. Program at KMUTT [grant number 39/2560].