Fast hybrid method for solving variational inequalities beyond monotonicity in real Banach spaces

Abstract

In this work, we introduce a new inertial hybrid algorithm that combines the subgradient extragradient, Halpern, and S-iteration methods. We provide an approximate solution to variational inequality and common fixed point problem of relatively non-expansive mappings in real Banach spaces, where our cost operator is Lipschitz continuous and quasimonotone, and our step size is self-adaptive, non-monotone, and non-reliant on any line-search principle. We provide two strong convergence results with (and without) recourse to the monotonicity property, and further conduct several relevant numerical experiments to reveal the efficiency of our method over others in the literature.

Keywords:

• Halpern
• quasimonotone
• self-adaptive step size
• subgradient extragradient
• S-iteration
• non-Lipschitz variational inequalities
• inertial iteration
• strong convergence

2020 Mathematics Subject Classifications:

• 65K15
• 65J15
• 47J20
• 90C25
• 90C33

Acknowledgements

The authors sincerely thank the Associate Editor and anonymous referees for their careful reading, constructive comments and useful suggestions that improved the manuscript. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

Disclosure statement

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

Data availability statement

Not applicable.

Additional information

Funding

The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers [grant number 119903].