Abstract
It is known that several optimization problems can be converted to a fixed point problem for which the underline fixed point operator is an averaged quasi-nonexpansive mapping and thus the corresponding fixed point method utilizes to solve the considered optimization problem. In this paper, we consider a fixed point method involving inertial extrapolation step with relaxation parameter to obtain a common fixed point of a countable family of averaged quasi-nonexpansive mappings in real Hilbert spaces. Our results bring a unification of several versions of fixed point methods for averaged quasi-nonexpansive mappings considered in the literature and give several implications of our results. We also give some applications to monotone inclusion problem with three-operator splitting method and composite convex and non-convex relaxed inertial proximal methods to solve both convex and nonconvex reweighted \(l_Q\) regularization for recovering a sparse signal. Finally, some numerical experiments are drawn from sparse signal recovery to illustrate our theoretical results.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Alvarez F (2003) Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J Optim 14:773–782
Alvarez F, Attouch H (2001) An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Var Anal 9:3–11
Attouch H, Czarnecki MO (2002) Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J Differ Equ 179(1):278–310
Attouch H, Goudon X, Redont P (2000) The heavy ball with friction. I. The continuous dynamical system. Commun Contemp Math 2(1):1–34
Attouch H, Bolte J, Svaiter BF (2013) Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math Program Ser A 137(1–2):91–129
Bauschke HH, Combettes PL (2011) Convex analysis and monotone operator theory in Hilbert spaces. CMS Books in Mathematics. Springer, New York
Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2(1):183–202
Berinde V (2007) Iterative approximation of fixed points, vol 1912. Lecture Notes in Mathematics. Springer, Berlin
Borwein JM, Li G, Tam MK (2017) Convergence rate analysis for averaged fixed point iterations in common fixed point problems. SIAM J Optim 27:1–33
Boţ RI, Csetnek ER, Hendrich C (2015) Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl Math Comput 256:472–487
Cai X, Gu G, He B (2014) On the \(O(1/t)\) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput Optim Appl 57:339–363
Candès E, Romberg J, Tao T (2006) Robust uncertainty principles: Exact signal reconstruction from highly incomplete Fourier information. IEEE Trans Inform Theory 52:489–509
Cegielski A (2012) Iterative methods for fixed point problems in Hilbert spaces, vol 2057. Lecture Notes in Mathematics. Springer, Berlin
Chang SS, Cho YJ, Zhou H (eds) (2002) Iterative Methods for Nonlinear Operator Equations in Banach Spaces. Nova Science, Huntington, NY
Chuang C-S, Takahashi W (2015) Weak convergence theorems for families of nonlinear mappings with generalized parameters. Numer Funct Anal Optim 36:41–54
Corman E, Yuan X (2014) A generalized proximal point algorithm and its convergence rate estimate. SIAM J Optim 24:1614–1638
Damek D, Watao Y (2017) A three-operator splitting scheme and its optimization applications. Set-Valued Var Anal 25:829–858
Dong Y (2015) Comments on “the proximal point algorithm revisited”. J Optim Theory Appl 116:343–349
Donoho DL (2006) Compressed sensing. IEEE Trans Inform Theory 52(4):1289–1306
He BS (1997) A class of projection and contraction methods for monotone variational inequalities. Appl Math Optim 35:69–76
He BS, Yuan XM (2015) On the convergence rate of Douglas–Rachford operator splitting method. Math Program 153:715–722
Jules F, Maingé PE (2002) Numerical approaches to a stationary solution of a second order dissipative dynamical system. Optimization 51:235–255
Maingé PE (2007) Inertial iterative process for fixed points of certain quasi-nonexpansive mappings. Set-Valued Var Anal 15:67–79
Maingé PE (2008a) Regularized and inertial algorithms for common fixed points of nonlinear operators. J Math Anal Appl 344:876–887
Maingé PE (2008b) Convergence theorems for inertial KM-type algorithms. J Comput Appl Math 219(1):223–236
Opial Z (1967) Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc 73:591–597
Peng B, Xu H-K (2020) Proximal methods for reweighted \(l_Q\)-regularization of sparse signal recovery. Appl Math Comput 386:125408
Polyak BT (1964) Some methods of speeding up the convergence of iterarive methods. Zh Vychisl Mat Mat Fiz 4:1–17
Shehu Y (2018) Convergence Rate Analysis of Inertial Krasnoselskii-Mann-type Iteration with Applications. Numer Funct Anal Optim 39:1077–1091
Svaiter BF (2011) On weak convergence of the Douglas–Rachford method. SIAM J Control Optim 49:280–287
Voronin S, Daubechies I (2017) An iteratively reweighted least squares algorithm for sparse regularization, Functional analysis, harmonic analysis, and image processing: a collection of papers in honor of Bjórn Jawerth, 391–411, Contemp. Math., 693, Amer. Math. Soc., Providence, RI, 2017
Xu H-K (2011) Averaged mappings and the gradient-projection algorithm. J Optim Theory Appl 150:360–378
Acknowledgements
This paper is dedicated to the loving memory of late Professor Charles Ejike Chidume (1947–2021).
Funding
No funding was received for this manuscript.
Author information
Authors and Affiliations
Contributions
All the authors contributed equally to the final version of the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
All the authors approved the final version of this manuscript.
Informed consent
All the authors understand the purpose of the research and their individual role.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shehu, Y., Dong, QL., Hu, Z. et al. Relaxed inertial fixed point method for infinite family of averaged quasi-nonexpansive mapping with applications to sparse signal recovery. Soft Comput 26, 1793–1809 (2022). https://doi.org/10.1007/s00500-021-06416-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-021-06416-7