An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions

Abstract

In this paper, we propose a new inertial extrapolation method for solving the generalized split feasibility problems over the solution set of monotone variational inclusion problems in real Hilbert spaces. We prove that the proposed method converges strongly to a solution to the aforementioned problem under the assumption that the associated singlevalued operator for the monotone variational inclusion problem is monotone and Lipschitz continuous. Our method uses a stepsize that is generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the Lipschitz constant of the singlevalued operator. Moreover, we discuss some consequences of our results and apply them to solve the split linear inverse problems, for which we also considered a special case of the split linear inverse problem, namely, the LASSO problem. We also give some numerical illustrations of the proposed method in comparison with other method in the literature to further demonstrate the applicability and efficiency of our method.

Keywords:

• Generalized split feasibility problems
• variational inequalities
• monotone variational inclusions
• inertial method
• linear inverse problem

2010 Mathematics Subject Classifications:

• 47H09
• 47H10
• 49J20
• 49J40

Acknowledgments

The authors sincerely thank the anonymous referees for their careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The research of the first author is wholly supported by the National Research Foundation (NRF) South Africa (S& F-DSI/NRF Free Standing Postdoctoral Fellowship; Grant Number: 120784). The first author also acknowledges the financial support from DSI/NRF, South Africa Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Postdoctoral Fellowship. The third author is supported by the NRF of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to NRF or CoE-MaSS.

Disclosure statement

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

Additional information

Funding

The research of the first author is wholly supported by the National Research Foundation (NRF) South Africa (S& F-DSI/NRF Free Standing Postdoctoral Fellowship) [grant number 120784]. The first author also acknowledges the financial support from DSI/NRF, South Africa Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Postdoctoral Fellowship. The third author is supported by the NRF of South Africa Incentive Funding for Rated Researchers [grant number 119903].