Regularization extragradient methods for equilibrium programming in Hilbert spaces

ABSTRACT

The paper introduces two new numerical methods for solving a variational inequality problem whose constraint set is expressed as the solution set of a monotone and Lipschitz-type equilibrium problem in a Hilbert space. We present how to combine regularization terms in an extragradient method and prove that the iterative sequences generated by the resulting methods converge strongly to a solution of equilibrium problem which solves the associated variational inequality problem. Theorems of strong convergence are analysed which are based on the incorporated Tikhonov regularization method for equilibrium problems. The first method is designed in the case where the Lipschitz-type constants of bifunction are known. While the second method can be implemented more easily without the prior knownledge of Lipschitz-type constants. The reason is that the second method have used a new stepsize rule whose computation is simple and easy to check at each step. Several numerical experiments are performed and they have demonstrated the effectiveness and the fast convergence of the new methods over existing methods.

Keywords:

• Equilibrium problem
• monotonicity
• Lipschitz condition
• extragradient method
• regularization

2010 Mathematics Subject Classifications:

• 65K10
• 65K15
• 90C30
• 47H05

Acknowledgments

The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The first two authors have been supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2020.06.

Disclosure statement

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

Additional information

Funding

The first two authors have been supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.01-2020.06].