Annals of Mathematical Sciences and Applications

Volume 8 (2023)

Number 2

Special issue dedicated to Anthony To-Ming Lau on his 80th birthday

Guest Editors: Xiaolong Qin, Ngai-Ching Wong and Jen-Chih Yao

An alternated inertial algorithm with weak and linear convergence for solving monotone inclusions

Pages: 321 – 345



Bing Tan (School of Sciences, Southwest Petroleum University, Chengdu, China; and Department of Mathematics, University of British Columbia, Kelowna, BC, Canada)

Xiaolong Qin (Department of Mathematics, Hangzhou Normal University, Hangzhou, China)


Inertial-based methods have the drawback of not preserving the Fejér monotonicity of iterative sequences, which may result in slower convergence compared to their corresponding non-inertial versions. To overcome this issue, Mu and Peng [Stat. Optim. Inf. Comput. 3 (2015), 241–248; $\href{}{MR3393305}$] suggested an alternating inertial method that can recover the Fejér monotonicity of even subsequences. In this paper, we propose a modified version of the forward-backward algorithm with alternating inertial and relaxation effects to solve an inclusion problem in real Hilbert spaces. The weak and linear convergence of the presented algorithm is established under suitable and mild conditions on the involved operators and parameters. Furthermore, the Fejér monotonicity of even subsequences generated by the proposed algorithm with respect to the solution set is recovered. Finally, our tests on image restoration problems demonstrate the superiority of the proposed algorithm over some related results.


inclusion problems, alternated inertial, forward-backward method, Tseng’s method, projection and contraction method, linear convergence

2010 Mathematics Subject Classification

Primary 47J20, 49J40, 65K15, 68W10. Secondary 90C33.

Dedicated to Professor Anthony To-Ming Lau on the occasion of his 80th birthday

Bing Tan is supported by the China Scholarship Council (CSC No. 202106070094).

Accepted 25 May 2023

Published 26 July 2023