Modified extragradient method with Bregman distance for variational inequalities

ABSTRACT

The paper deals with a numerical method for solving a monotone variational inequality problem in a Hilbert space. The algorithm is inspired by Popov's modified extragradient method and the Bregman projection with a simple stepsize rule. Applying Bregman projection allows the algorithm to be more flexible in computations when choosing a projection. The stepsizes, which vary from step to step, are found over each iteration by a cheap computation without any linesearch. The convergence of the algorithm is proved without the prior knowledge of Lipschitz constant of the operator involved. Some numerical experiments are performed to illustrate the computational performances of the new algorithm with several known Bregman distances. The obtained results in this paper extend some existing results in the literature.

Keywords:

• Variational inequality
• monotone operator
• extragradient method
• Bregman projection

2010 Mathematics Subject Classifications:

• 65Y05
• 65K15
• 68W10
• 47H05
• 47H10

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.

Disclosure statement

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

Additional information

Funding

The first author is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 101.01-2020.06. P. Cholamjiak was supported by Thailand Research Fund and University of Phayao under the project RSA6180084 and UOE62001. This work was partially supported by Thailand Science Research and Innovation grant no. IRN62W0007.