Global convergence in a modified RMIL-type conjugate gradient algorithm for nonlinear systems of equations and signal recovery

Yan Xia; Songhua Wang; Yan Xia; Songhua Wang

doi:10.3934/era.2024286

Electronic Research Archive

2024, Volume 32, Issue 11: 6153-6174. doi: 10.3934/era.2024286

Previous Article Next Article

Research article Topical Sections

Global convergence in a modified RMIL-type conjugate gradient algorithm for nonlinear systems of equations and signal recovery

Yan Xia ¹,
Songhua Wang ^{2
,
,}

1.
School of Artificial Intelligence, Guangzhou Huashang College, Guangzhou 511300, China
2.
School of Mathematics, Physics and Statistics, Baise University, Baise 533099, China

Received: 06 September 2024 Revised: 20 October 2024 Accepted: 06 November 2024 Published: 15 November 2024

This paper proposes a modified Rivaie-Mohd-Ismail-Leong (RMIL)-type conjugate gradient algorithm for solving nonlinear systems of equations with convex constraints. The proposed algorithm offers several key characteristics: (1) The modified conjugate parameter is non-negative, thereby enhancing the proposed algorithm's stability. (2) The search direction satisfies sufficient descent and trust region properties without relying on any line search technique. (3) The global convergence of the proposed algorithm is established under general assumptions without requiring the Lipschitz continuity condition for nonlinear systems of equations. (4) Numerical experiments indicated that the proposed algorithm surpasses existing similar algorithms in both efficiency and stability, particularly when applied to large scale nonlinear systems of equations and signal recovery problems in compressed sensing.
- nonlinear systems of equations,
- conjugate gradient method,
- large scale,
- global convergence,
- signal recovery
Citation: Yan Xia, Songhua Wang. Global convergence in a modified RMIL-type conjugate gradient algorithm for nonlinear systems of equations and signal recovery[J]. Electronic Research Archive, 2024, 32(11): 6153-6174. doi: 10.3934/era.2024286

Related Papers:

Abstract

This paper proposes a modified Rivaie-Mohd-Ismail-Leong (RMIL)-type conjugate gradient algorithm for solving nonlinear systems of equations with convex constraints. The proposed algorithm offers several key characteristics: (1) The modified conjugate parameter is non-negative, thereby enhancing the proposed algorithm's stability. (2) The search direction satisfies sufficient descent and trust region properties without relying on any line search technique. (3) The global convergence of the proposed algorithm is established under general assumptions without requiring the Lipschitz continuity condition for nonlinear systems of equations. (4) Numerical experiments indicated that the proposed algorithm surpasses existing similar algorithms in both efficiency and stability, particularly when applied to large scale nonlinear systems of equations and signal recovery problems in compressed sensing.

References

[1]	M. Sun, Y. Wang, General five-step discrete-time Zhang neural network for time-varying nonlinear optimization, Bull. Malays. Math. Sci. Soc., 43 (2020), 1741–1760. https://doi.org/10.1007/s40840-019-00770-4 doi: 10.1007/s40840-019-00770-4
[2]	K. Meintjes, A. P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333–361. https://doi.org/10.1016/0096-3003(87)90076-2 doi: 10.1016/0096-3003(87)90076-2
[3]	D. Li, S. Wang, Y. Li, J. Wu, A projection-based hybrid PRP-DY type conjugate gradient algorithm for constrained nonlinear equations with applications, Appl. Numer. Math., 195 (2024), 105–125. https://doi.org/10.1016/j.apnum.2023.09.009 doi: 10.1016/j.apnum.2023.09.009
[4]	D. Li, J. Wu, Y. Li, S. Wang, A modified spectral gradient projection-based algorithm for large-scale constrained nonlinear equations with applications in compressive sensing, J. Comput. Appl. Math., 424 (2023), 115006. https://doi.org/10.1016/j.cam.2022.115006 doi: 10.1016/j.cam.2022.115006
[5]	M. W. Yusuf, L. W. June, M. A. Hassan, Jacobian-free diagonal Newton's method for solving nonlinear systems with singular Jacobian, Malays. J. Math. Sci., 5 (2011), 241–255.
[6]	Q. Yan, X. Peng, D. Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations, J. Comput. Appl. Math., 234 (2010), 649–657. https://doi.org/10.1016/j.cam.2010.01.001 doi: 10.1016/j.cam.2010.01.001
[7]	H. Abdullahi, A. S. Halilu, M. Y. Waziri, A modified conjugate gradient method via a double direction approach for solving large-scale symmetric nonlinear equations, J. Numer. Math. Stoch., 10 (2018), 32–44.
[8]	I. Yusuf, A. S. Halilu, M. Y. Waziri, Efficient matrix-free direction method with line search for solving large scale systems of nonlinear equations, Yugosl. J. Oper. Res., 30 (2020), 399–412. https://doi.org/10.2298/YJOR160515005H doi: 10.2298/YJOR160515005H
[9]	D. Q. Huynh, F. N. Hwang, An accelerated structured quasi-Newton method with a diagonal second-order Hessian approximation for nonlinear least squares problems, J. Comput. Appl. Math., 442 (2024), 115718. https://doi.org/10.1016/j.cam.2023.115718 doi: 10.1016/j.cam.2023.115718
[10]	X. Wu, H. Shao, P. Liu, An efficient conjugate gradient-based algorithm for unconstrained optimization and its projection extension to large-scale constrained nonlinear equations with applications in signal recovery and image denoising problems, J. Comput. Appl. Math., 422 (2023), 114879. https://doi.org/10.1016/j.cam.2022.114879 doi: 10.1016/j.cam.2022.114879
[11]	G. Ma, J. Jiang, J. Jian, A modified inertial three-term conjugate gradient projection method for constrained nonlinear equations with applications in compressed sensing, Numer. Algor., 92 (2023), 1621–1653. https://doi.org/10.1007/s11075-022-01356-1 doi: 10.1007/s11075-022-01356-1
[12]	W. Liu, J. Jian, J. Yin, An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations, Numer. Algor., 97 (2024), 985–1015. https://doi.org/10.1007/s11075-023-01736-1 doi: 10.1007/s11075-023-01736-1
[13]	S. B. Salihu, A. S. Halilu, M. Abdullahi, An improved spectral conjugate gradient projection method for monotone nonlinear equations with application, J. Appl. Math. Comput., 70 (2024), 3879–3915. https://doi.org/10.1007/s12190-024-02121-4 doi: 10.1007/s12190-024-02121-4
[14]	Y. Narushima, H. Yabe, J. A. Ford, A three-term conjugate gradient method with sufficient descent property for unconstrained optimization, SIAM J. Optim., 21 (2011), 212–230. https://doi.org/10.1137/080743573 doi: 10.1137/080743573
[15]	Y. Narushima, A smoothing conjugate gradient method for solving systems of nonsmooth equations, Appl. Math. Comput., 219 (2013), 8646–8655. https://doi.org/10.1016/j.amc.2013.02.060 doi: 10.1016/j.amc.2013.02.060
[16]	R. Huang, Y. Qin, K. Liu, G. Yuan, Biased stochastic conjugate gradient algorithm with adaptive step size for nonconvex problems, Expert Syst. Appl., 238 (2024), 121556. https://doi.org/10.1016/j.eswa.2023.121556 doi: 10.1016/j.eswa.2023.121556
[17]	X. Jiang, Y. Zhu, J. Jian, Two efficient nonlinear conjugate gradient methods with restart procedures and their applications in image restoration, Nonlinear Dyn., 111 (2023), 5469–5498. https://doi.org/10.1007/s11071-022-08013-1 doi: 10.1007/s11071-022-08013-1
[18]	W. Cheng, A PRP type method for systems of monotone equations, Math. Comput. Model., 50 (2009), 15–20. https://doi.org/10.1016/j.mcm.2009.04.007 doi: 10.1016/j.mcm.2009.04.007
[19]	G. Yu, A derivative-free method for solving large-scale nonlinear systems of equations, J. Ind. Manag. Optim., 6 (2009), 149–160. https://doi.org/10.3934/jimo.2010.6.149 doi: 10.3934/jimo.2010.6.149
[20]	M. Y. Waziri, K. Ahmed, J. Sabi'u, A family of Hager-Zhang conjugate gradient methods for system of monotone nonlinear equations, Appl. Math. Comput., 361 (2019), 645–660. https://doi.org/10.1016/j.amc.2019.06.012 doi: 10.1016/j.amc.2019.06.012
[21]	P. Liu, H. Shao, Z. Yuan, T. Zheng, A family of three-term conjugate gradient projection methods with a restart procedure and their relaxed-inertial extensions for the constrained nonlinear pseudo-monotone equations with applications, Numer. Algor., 94 (2023), 1055–1083. https://doi.org/10.1007/s11075-023-01527-8 doi: 10.1007/s11075-023-01527-8
[22]	A. Ibrahim, M. Alshahrani, S. Al-Homidan, Two classes of spectral three-term derivative-free method for solving nonlinear equations with application, Numer. Algor., 96 (2024), 1625–1645. https://doi.org/10.1007/s11075-023-01679-7 doi: 10.1007/s11075-023-01679-7
[23]	M. Rivaie, M. Mamat, L. W. June, I. Mohd, A new class of nonlinear conjugate gradient coefficients with global convergence properties, Appl. Math. Comput., 218 (2012), 11323–11332. https://doi.org/10.1016/j.amc.2012.05.030 doi: 10.1016/j.amc.2012.05.030
[24]	Z. Dai, Comments on a new class of nonlinear conjugate gradient coefficients with global convergence properties, Appl. Math. Comput., 276 (2016), 297–300. https://doi.org/10.1016/j.amc.2015.11.085 doi: 10.1016/j.amc.2015.11.085
[25]	A. B. Abubakar, P. Kumam, H. Mohammad, A modified Fletcher-Reeves conjugate gradient method for monotone nonlinear equations with some applications, Mathematics, 7 (2019), 745. https://doi.org/10.3390/math7080745 doi: 10.3390/math7080745
[26]	J. Yin, J. Jian, X. Jiang, M. Liu, L. Wang, A hybrid three-term conjugate gradient projection method for constrained nonlinear monotone equations with applications, Numer. Algor., 88 (2021), 389–418. https://doi.org/10.1007/s11075-020-01043-z doi: 10.1007/s11075-020-01043-z
[27]	E. D. Dolan, J. Jorge, Benchmarking optimization software with performance profiles, Math. Program., 91 (2001), 201–213. https://doi.org/10.1007/s101070100263 doi: 10.1007/s101070100263
[28]	D. Li, S. Wang, Y. Li, J. Wu, A convergence analysis of hybrid gradient projection algorithm for constrained nonlinear equations with applications in compressed sensing, Numer. Algor., 95 (2024), 1325–1345. https://doi.org/10.1007/s11075-023-01610-0 doi: 10.1007/s11075-023-01610-0

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)