New extragradient method for a class of equilibrium problems in Hilbert spaces

Abstract

The paper proposes a new extragradient algorithm for solving strongly pseudomonotone equilibrium problems which satisfy a Lipschitz-type condition recently introduced by Mastroeni in auxiliary problem principle. The main novelty of the paper is that the algorithm generates the strongly convergent sequences in Hilbert spaces without the prior knowledge of Lipschitz-type constants and any hybrid method. Several numerical experiments on a test problem are also presented to illustrate the convergence of the algorithm.

Keywords:

• Equilibrium problem
• extragradient method
• pseudomonotone bifunction
• monotone bifunction

AMS Subject Classifications:

• 90C33
• 47J20

Acknowledgements

The author would like to thank the Associate Editor and two anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The guidance of Profs. P. K. Anh and L. D. Muu is gratefully acknowledged.

Notes

No potential conflict of interest was reported by the author.

1 We randomly choose . We set , as two diagonal matrixes with eigenvalues and , respectively. Then, we construct a positive semidefinite matrix Q and a negative definite matrix T by using random orthogonal matrixes with and , respectively. Finally, we set