Sparse signal reconstruction via Hager–Zhang-type schemes for constrained system of nonlinear equations

Abstract

In this article, two Hager–Zhang (HZ) type projection algorithms are presented for large-dimension nonlinear monotone problems and sparse signal recovery in compressed sensing. This goal is attained by conducting singular value analysis of a nonsingular HZ-type search direction matrix as well as applying the idea by Piazza and Politi [J Comput Appl Math. 2002;143(1):141–144] and minimizing the Frobenius norm of an orthornormal matrix. The paper attempts to fill the gap in the work of Hager and Zhang [Pac J Optim. 2006;2(1):35–58], Waziri et al. [Appl Math Comput. 2019;361:645–660], Sabi'u et al. [Appl Numer Math. 2020;153:217–233] and Babaie-Kafaki [4OR-Q J Oper Res. 2014;12:285-292], where the sufficient descent or global convergence condition is not satisfied when the HZ parameter is in the interval $(0,\frac{1}{4})$. The proposed schemes are also suitable for solving non-smooth nonlinear problems. Also, by employing some mild conditions, global convergence of the schemes are established, while numerical comparison with four effective HZ-type methods show that the new methods are efficient. Furthermore, to illustrate their practical application, both methods are applied to solve the ${\ell }_1$-norm regularization problems to recover a sparse signal in compressive sensing. The experiments conducted in that regard show that the methods are promising and perform better than two other methods in the literature.

Keywords:

• Nonlinear monotone equations
• line search
• projection method
• convex constraint
• compressed sensing

Mathematics Subject Classifications:

• 90C30
• 90C26

Acknowledgments

We are grateful to the anonymous reviewers and associate editor for all their comments and suggestions that has improved the quality of the work. We also thank the members of the Numerical optimization research group, Bayero university, Kano for their advise and support.

Disclosure statement

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