Three-operator reflected forward-backward splitting algorithm with double inertial effects

Abstract

In this paper, we propose a reflected forward-backward splitting algorithm with two different inertial extrapolations to find a zero of the sum of three monotone operators consisting of the maximal monotone operator, Lipschitz continuous monotone operator, and a cocoercive operator in real Hilbert spaces. One of the interesting features of the proposed algorithm is that both the Lipschitz continuous monotone operator and the cocoercive operator are computed explicitly each with one evaluation per iteration. We then obtain weak and strong convergence results under some mild conditions. We finally give a numerical illustration to show that our proposed algorithm is effective and competitive with other related algorithms in the literature.

KEYWORDS:

• Three-operator splitting algorithm
• inertial extrapolation step
• weak and strong convergence

2010 MSC CLASSIFICATIONS:

• 90C25
• 90C30
• 90C60
• 68Q25
• 49M25

Acknowledgments

The authors are grateful to the associate editor and the two anonymous referees for their excellent suggestions and comments which have improved the earlier version of our manuscript greatly.

Disclosure statement

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

Availability of data and material

Not applicable.

Code availability

The Matlab codes employed to run the numerical experiments are available at https://github.com/suminmath/codes.

Additional information

Funding

The second author was supported by the Science and Technology Innovation Project for Graduate Students of Tianjin [grant number 2022SKY165].

Notes on contributors

Qiao-Li Dong

Qiao-Li Dong is a Professor, at the College of Science, Civil Aviation University of China. Her research Interest are Nonlinear Analysis and Optimization Theory and Methods, including variational inequalities, fixed point methods and splitting methods as well as the inertial accelerations.

Min Su

Min Su is a graduate student, at the College of Science, Civil Aviation University of China, Tianjin, People's Republic of China. Her research focuses on iterative algorithm for the nonlinear operator fixed point problem and participates in the Science and Technology Innovation Project for Graduate.

Yekini Shehu

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