Two Bregman projection methods for solving variational inequalities

ABSTRACT

Using Bregman distances, we propose two extragradient-like methods for solving variational inequality problems with Lipschitz cost operators in a Hilbert space. Weak and strong convergence theorems for our algorithms are established when the cost operator is either monotone or pseudomonotone. The variable stepsizes are generated by the algorithms at each iterative stage without any line search procedure. Our stepsize rule allows the algorithms to be easily implemented without prior knowledge of the Lipschitz constant of the cost operator. We also provide several numerical findings in order to illustrate our theoretical results.

Keywords:

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

AMS Subject Classification::

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

Acknowledgments

This first author has been supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2020.06. The second author has been partially supported by the Israel Science Foundation (Grant No. 820/17), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. Both authors are grateful to an anonymous referee for several useful comments and helpful suggestions.

Disclosure statement

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

Additional information

Funding

This first author was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant no. 101.01-2020.06]. The second author was partially supported by the Israel Science Foundation [grant no. 820/17], by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.