A proximal point method for quasi-equilibrium problems in Hilbert spaces

Abstract

In this paper, we study the convergence of a proximal point method for solving quasi-equilibrium problems (QEP) in Hilbert spaces. We extent the method proposed by Moudafi [Proximal point algorithm extended to equilibrium problems. J Nat Geom. 1999;15(1-2):91–100] and Iusem and Sosa [Iterative algorithms for equilibrium problems. Optimization. 2003;52(3):301–316] to the more general context of quasi-equilibrium problems. In our method a quasi-equilibrium problem is solved by computing a solution of an equilibrium problem at each iteration. We obtain weak convergence of the sequence to a solution of the QEP under some mild assumptions. Some encouraging numerical experiments are presented to show the performance of the method.

Keywords:

• Quasi-equilibrium problem
• equilibrium problem
• proximal point method
• monotone bifunction

2010 Mathematics Subject Classifications:

• 49M37
• 65K15
• 90C33

Acknowledgments

This work is dedicated to Dr Alfredo Noel Iusem in honor to his 70th birthday. The authors wish to thank Dr Susana Scheimberg for her comments in first version of this paper. We also thank the associate editor and the referees for their constructive remarks which allowed us to improve the paper.

Disclosure statement

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

Additional information

Funding

The second author is supported in part by CNPq grant 424169/2018-5.