The subgradient extragradient method extended to equilibrium problems

Abstract

A globally convergent algorithm for equilibrium problems with pseudomonotone bifunctions is proposed. The algorithm is based on the idea of the subgradient extragradient method for solving variational inequalities proposed by Censor et al. [Y. Censor, A. Gibali, and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl. 148 (2011), 318–335.] and Armijo linesearch techniques. In addition, we give a modified version of our algorithm for finding a common point of the solution set of equilibrium problems and the fixed point set of a nonexpansive mapping. We also analyse the weak convergence of both algorithms in a real Hilbert space.

Keywords:

• equilibrium problems
• extragradient method
• subgradient
• pseudomonotonicity

AMS Subject Classifications::

• 65 K10
• 65 K15
• 90 C25
• 90 C33

Acknowledgements

We are very grateful to the anonymous referees for their really helpful and constructive comments that helped us very much to improve our article. The work presented here was completed while the first author was on leave at LITA, University of Lorraine, France. He wishes to thank the Fonds Européens de Développement Régional Lorraine for the financial support via the FEDER project ‘INNOMAD’.