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Volume 12
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Article

A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems

by
Aliyu Muhammed Awwal
1,2,
Lin Wang
3,*,
Poom Kumam
1,4,* and
Hassan Mohammad
5
1
KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
2
Department of Mathematics, Faculty of Science, Gombe State University, Gombe 760214, Nigeria
3
Office of Science and Research, Yunnan University of Finance and Economics, Kunming 650221, China
4
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5
Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano 700241, Nigeria
*
Authors to whom correspondence should be addressed.
Symmetry 2020, 12(6), 874; https://doi.org/10.3390/sym12060874
Submission received: 23 March 2020 / Revised: 15 April 2020 / Accepted: 16 April 2020 / Published: 26 May 2020
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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Abstract

In this paper, we propose a two-step iterative algorithm based on projection technique for solving system of monotone nonlinear equations with convex constraints. The proposed two-step algorithm uses two search directions which are defined using the well-known Barzilai and Borwein (BB) spectral parameters.The BB spectral parameters can be viewed as the approximations of Jacobians with scalar multiple of identity matrices. If the Jacobians are close to symmetric matrices with clustered eigenvalues then the BB parameters are expected to behave nicely. We present a new line search technique for generating the separating hyperplane projection step of Solodov and Svaiter (1998) that generalizes the one used in most of the existing literature. We establish the convergence result of the algorithm under some suitable assumptions. Preliminary numerical experiments demonstrate the efficiency and computational advantage of the algorithm over some existing algorithms designed for solving similar problems. Finally, we apply the proposed algorithm to solve image deblurring problem.
Keywords: spectral gradient method; nonlinear monotone equations; projection method; line search; image deblurring spectral gradient method; nonlinear monotone equations; projection method; line search; image deblurring

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MDPI and ACS Style

Awwal, A.M.; Wang, L.; Kumam, P.; Mohammad, H. A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems. Symmetry 2020, 12, 874. https://doi.org/10.3390/sym12060874

AMA Style

Awwal AM, Wang L, Kumam P, Mohammad H. A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems. Symmetry. 2020; 12(6):874. https://doi.org/10.3390/sym12060874

Chicago/Turabian Style

Awwal, Aliyu Muhammed, Lin Wang, Poom Kumam, and Hassan Mohammad. 2020. "A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems" Symmetry 12, no. 6: 874. https://doi.org/10.3390/sym12060874

APA Style

Awwal, A. M., Wang, L., Kumam, P., & Mohammad, H. (2020). A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems. Symmetry, 12(6), 874. https://doi.org/10.3390/sym12060874

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