Weak and strong convergence results for solving monotone variational inequalities in reflexive Banach spaces

Abstract

In this paper, we introduce two modified Tseng's extragradient algorithms with a new generalized adaptive stepsize for solving monotone variational inequalities (VI) in reflexive Banach spaces. The advantage of our methods is that stepsizes do not require prior knowledge of the Lipschitz constant of the cost mapping. Based on Bregman projection-type methods, we prove weak and strong convergence of the proposed algorithms to a solution of VI. Some numerical experiments to show the efficiency of our methods including a comparison with related methods are provided.

Keywords:

• Legendre function
• reflexive banach space
• weak convergence
• strong convergence
• Bregman projection

2010 Mathematics Subject Classifications:

• 47H09
• 47H10
• 47J25
• 47J05

Acknowledgements

The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped to improve the quality of this paper.

Disclosure statement

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

Additional information

Funding

P. Cholamjiak was supported by University of Phayao and Thailand Science Research and Innovation [grant number FF65-UoE001] and P. Sunthrayuth was supported by Rajamangala University of Technology Thanyaburi (RMUTT).