One-step Bregman projection methods for solving variational inequalities in reflexive Banach spaces

ABSTRACT

We study two Bregman projection methods for solving variational inequality problems in real reflexive Banach spaces. Our methods have simple and elegant structures, and they require only one Bregman projection onto the feasible set and one evaluation of the cost operator at each iteration. We prove that these methods converge weakly when the cost operator is pseudomonotone on the entire space and that they converge strongly when the cost operator is strongly pseudomonotone only on the feasible set. Extensions of our proposed methods to mixed variational inequality problems are also given. Finally, we consider some examples regarding the implementation of our methods in comparisons with known methods in the literature.

KEYWORDS:

• Bregman distance
• Bregman projection
• pseudomonotone operator
• reflexive Banach space
• variational inequalities

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 47H09
• 47H10
• 49J20
• 49J40

Acknowledgments

The authors are grateful to the associate editor and the two anonymous referees for their insightful comments and suggestions which have improved greatly on the earlier version of the paper.

Disclosure statement

The authors declare that they have no conflict of interest.

Code availability

The Matlab codes employed to run the numerical experiments are available upon request to the authors.

Additional information

Funding

The second author was partially supported by the Israel Science Foundation (Grant 820/17), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.