A new inertial projected reflected gradient method with application to optimal control problems

Abstract

The projected reflected gradient method has been shown to be a simple and elegant method for solving variational inequalities. The method involves one projection onto the feasible set and one evaluation of the cost operator per iteration and has been shown numerically to be more efficient than most available methods for solving variational inequalities. Convergence results for methods with similar elegant structures of projected reflected gradient method are still rare. In this paper, we present weak and linear convergence of a projected reflected gradient method with an inertial extrapolation step and give some applications arising from optimal control problems. We first obtain weak convergence result for the projected reflected gradient method with an inertial extrapolation step for solving variational inequalities under standard assumptions with self-adaptive step sizes. We further obtain a linear convergence rate when the cost operator is strongly monotone and Lipschitz continuous. Finally, we give some numerical applications arising from optimal control. Preliminary results show that our method is effective and efficient when compared to other related state-of-the-art methods in the literature and show the advantage gained by incorporating inertial terms into the projected reflected gradient methods.

Keywords:

• Variational inequalities
• inertial projection method
• weak convergence
• linear convergence
• optimal control
• Hilbert spaces

2020 MSC classifications:

• 47H05
• 47J20
• 47J25
• 65K15
• 90C25

Acknowledgments

We are very thankful to the referee and Senior Editor for the valuable comments and suggestions that helped to improve our manuscript.

Declarations

Code availability

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

Disclosure statement

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

Additional information

Notes on contributors

Chinedu Izuchukwu

Chinedu Izuchukwu is a Lecturer at the School of Mathematics, University of the Witwatersrand, South Africa. He obtained his doctoral degree from the University of KwaZulu-Natal, South Africa. He was a postdoctoral fellow at the Technion-Israel Institute of Technology, Haifa, Israel. His research interests lie in the areas of continuous and discrete optimization.

Yekini Shehu

Yekini Shehu is a Professor of Mathematics at the School of Mathematical Sciences, Zhejiang Normal University, Jinhua, China. His research include Optimization & Variational Problems, Bilevel Optimization Problems, Quasi-variational Inequalities, Nonlinear Integral Equations, Monotone Inclusion Problems and Variational Inequalities. Professor Shehu is a recipient of the prestigious Alexander von Humboldt Postdoctoral Research Fellowship awarded by the Alexander von Humboldt Foundation, Bonn, Germany, from 01 May 2016 to 30 April 2019. He was also a postdoctoral fellow at the Institute of Science and Technology (IST), Vienna, Austria from 01 June 2019 to 30 September 2021.