A Bregman subgradient extragradient method with self-adaptive technique for solving variational inequalities in reflexive Banach spaces

Abstract

In this paper, we introduce a self-adaptive Bregman subgradient extragradient method for solving variational inequalities in the framework of a reflexive Banach space. The step-adaptive strategy avoids the difficult task of choosing a stepsize based on the Lipschitz constant of the cost function of the variational inequalities and improves the performance of the algorithm. Moreover, the use of the Bregman distance technique allows the consideration of a general feasible set for the problem. Under some suitable conditions, we prove some weak and strong convergence results for the sequence generated by the algorithm without prior knowledge of the Lipschitz constant. We further provide an application to contact problems and some numerical experiments to illustrate the performance of the algorithm.

Keywords:

• Bregman distance
• variational inequalities
• self-adaptive stepsize
• extragradient method
• pseudomonotone operators

AMS Subject Classifications:

• 47H09
• 65K10
• 90C25
• 49J25

Acknowledgments

O.K. Oyewole acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. L.O. Jolaoso and K.O. Aremu thanks the Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Science University, Pretoria for making their facilities available for the research. L.O. Jolaoso is supported by the Postdoctoral research grant from the Sefako Makgatho Health Sciences University, South Africa.

Disclosure statement

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

Additional information

Funding

This research is funded by the Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Science University, Pretoria.