Accelerated Hager-Zhang type projection scheme for monotone equations with applications

Abubakar Sani Halilu; Mohamad A. Mohamed; Mohammed A. Saleh; Kabiru Ahmed; Abdulgader Z. Almaymuni; Mohammed Y. Waziri; Sulaiman M. Ibrahim; Badr Almutairi; Abubakar Sani Halilu; Mohamad A. Mohamed; Mohammed A. Saleh; Kabiru Ahmed; Abdulgader Z. Almaymuni; Mohammed Y. Waziri; Sulaiman M. Ibrahim; Badr Almutairi

doi:10.3934/math.20251238

AIMS Mathematics

2025, Volume 10, Issue 12: 28151-28181. doi: 10.3934/math.20251238

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Accelerated Hager-Zhang type projection scheme for monotone equations with applications

1.
Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Besut, Malaysia
2.
Numerical Optimization Research Group, Bayero University, Kano, Nigeria
3.
Mathematics Department, Sule Lamido University (SLU) Kafin Hausa, Nigeria
4.
Mathematical Innovation and Applications Research Group, SLU, Kafin Hausa, Nigeria
5.
Department of Cybersecurity, College of Computer, Qassim University, Saudi Arabia
6.
School of Quantitative Science, Universiti Utara Malaysia, Sintok, Malaysia
7.
Faculty of Education and Arts Sohar University, Sohar 311, Oman
8.
College of Computer and Information Sciences, Majmaah University, Almajmaah, Saudi Arabia

Received: 20 July 2025 Revised: 17 October 2025 Accepted: 11 November 2025 Published: 01 December 2025
MSC : 90C26, 90C30

This paper proposed a new iterative method for solving nonlinear monotone equations with convex constraint and its applications in sparse signal reconstruction and image de-blurring problems. The method can be viewed as an improved adaptation of the generalized Hager-Zhang conjugate gradient method for unconstrained optimization. Unlike the latter which only converged globally for strongly convex functions when the Hager-Zhang parameter $ \theta_k $ lies in the interval $ \left(\frac{1}{4}, +\infty\right) $, the new method exhibited this attribute for nonlinear monotone and Lipschitz continuous functions without restriction for $ \theta_k $ under a more relaxed condition. The derivative-free structure of the method made it suitable for both smooth and non-smooth problems. Numerical experiments on benchmark test problems demonstrated the method's superior performance compared to some state-of-the-art algorithms. Furthermore, the algorithm was successfully applied to sparse signal recovery and image de-blurring problems in compressed sensing, confirming its practical effectiveness.
- nonlinear monotone equations,
- convex constraint,
- conjugate gradient method,
- sparse signal reconstruction,
- image de-blurring,
- image reconstruction,
- optimization method
Citation: Abubakar Sani Halilu, Mohamad A. Mohamed, Mohammed A. Saleh, Kabiru Ahmed, Abdulgader Z. Almaymuni, Mohammed Y. Waziri, Sulaiman M. Ibrahim, Badr Almutairi. Accelerated Hager-Zhang type projection scheme for monotone equations with applications[J]. AIMS Mathematics, 2025, 10(12): 28151-28181. doi: 10.3934/math.20251238

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Abstract

This paper proposed a new iterative method for solving nonlinear monotone equations with convex constraint and its applications in sparse signal reconstruction and image de-blurring problems. The method can be viewed as an improved adaptation of the generalized Hager-Zhang conjugate gradient method for unconstrained optimization. Unlike the latter which only converged globally for strongly convex functions when the Hager-Zhang parameter $ \theta_k $ lies in the interval $ \left(\frac{1}{4}, +\infty\right) $, the new method exhibited this attribute for nonlinear monotone and Lipschitz continuous functions without restriction for $ \theta_k $ under a more relaxed condition. The derivative-free structure of the method made it suitable for both smooth and non-smooth problems. Numerical experiments on benchmark test problems demonstrated the method's superior performance compared to some state-of-the-art algorithms. Furthermore, the algorithm was successfully applied to sparse signal recovery and image de-blurring problems in compressed sensing, confirming its practical effectiveness.

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