MiKM: multi-step inertial Krasnosel’skiǐ–Mann algorithm and its applications

• Published in: Journal of global optimization Vol. 73; no. 4; pp. 801 - 824
• Main Authors: Dong, Q. L., Huang, J. Z., Li, X. H., Cho, Y. J., Rassias, Th. M.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2019; Springer Nature B.V
• Subjects: Algorithms; Computer Science; Convergence; Hilbert space; Iterative methods; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Real Functions; Splitting; Backward–forward splitting method; Nonexpansive operator; Monotone inclusion; Bounded perturbation resilience; Forward–backward splitting method; Multi-step inertial Krasnosel’skiǐ–Mann algorithm; Douglas–Rachford splitting method; Davis–Yin splitting method
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-018-0727-x

In this paper, we first introduce a multi-step inertial Krasnosel’skiǐ–Mann algorithm (MiKM) for nonexpansive operators in real Hilbert spaces. We give the convergence of the MiKM by investigating the convergence of the Krasnosel’skiǐ–Mann algorithm with perturbations. We also establish global pointwise and ergodic iteration complexity bounds of the Krasnosel’skiǐ–Mann algorithm with perturbations. Based on the MiKM, we construct some multi-step inertial splitting methods, including the multi-step inertial Douglas–Rachford splitting method (MiDRS), the multi-step inertial forward–backward splitting method, multi-step inertial backward–forward splitting method and and the multi-step inertial Davis–Yin splitting method. Numerical experiments are provided to illustrate the advantage of the MiDRS over the one-step inertial DRS and the original DRS.