Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems

Abstract

In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the setting of Hilbert space. We propose a modified inertial viscosity subgradient extragradient algorithm with self-adaptive stepsize in which the two projections are made onto some half-spaces. Moreover, we obtain a strong convergence result for approximating a common solution of the variational inequality and fixed point of quasi-nonexpansive mappings under some mild conditions. The main advantages of our method are: the self adaptive step-size which avoids the need to know apriori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which speeds up the rate of convergence of the algorithm. Numerical experiments are presented to demonstrate the efficiency of our algorithm in comparison with other existing algorithms in literature.

Keywords:

• Extragradient method
• inertia
• monotone
• variational inequality
• fixed point
• Lipschitz-continuous

2010 Mathematics Subject Classifications:

• 65K15
• 47J25
• 65J15
• 90C33

Acknowledgments

The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS and NRF.

Disclosure statement

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

Additional information

Funding

The second author 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. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903).