A Hager–Zhang Riemannian conjugate gradient method for matrix approximation

Nasiru Salihu; Seyed Yaser Mousavi Siamakani; Auwal Bala Abubakar; Also Mohammed Saleh; Nasiru Salihu; Seyed Yaser Mousavi Siamakani; Auwal Bala Abubakar; Also Mohammed Saleh

doi:10.3934/math.2026517

AIMS Mathematics

2026, Volume 11, Issue 5: 12580-12605. doi: 10.3934/math.2026517

Previous Article Next Article

Research article

A Hager–Zhang Riemannian conjugate gradient method for matrix approximation

1.
Department of Mathematics, Faculty of Sciences, Modibbo Adama University, Yola, 652105, Nigeria
2.
College of Engineering, Department of Civil Engineering, Rangsit University, Mueang, Pathum Thani, Thailand, 12000, Thailand
3.
Department of Art and Science, George Mason University, Songdomunhwa-ro 119-4, Yeonsu-gu, Incheon 21985, Republic of Korea
4.
Numerical Optimization Research Group, Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano 700241, Nigeria
5.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria 0204, Medunsa, South Africa
6.
Faculty of Education and Arts, Sohar University, Sohar 311, Oman
7.
College of Agriculture, Science and Technology, Jalingo, Nigeria

Received: 20 January 2026 Revised: 31 March 2026 Accepted: 01 April 2026 Published: 07 May 2026
MSC : 65K05, 90C30

We introduce a class of Hager–Zhang-type Riemannian conjugate gradient (CG) methods that generalize the framework of Sakai et al. (Applied Mathematics and Computation, 441 (2023) 127685) to arbitrary retractions while significantly extending its theoretical and practical scope. These methods ensure global convergence for non-convex problems without relying on strong convexity and inherently satisfy the sufficient descent property, independent of Riemannian line search conditions. A key algorithmic innovation is the introduction of an adaptive min(max) strategy to adjust the CG parameter $\beta_{k+1}$ using bounded parameters to ensure numerical stability. Furthermore, we extended classical Euclidean CG portfolio optimization to the sphere manifold, naturally enforcing budget constraints and improving robustness. Numerical experiments, available at GitHub repo, showed that our methods outperform classical Riemannian CG in iterations and computational time for large-scale problems.
- manifold optimization,
- Hager–Zhang-type CG methods,
- sufficient descent property,
- scaled vector transport,
- convergence analysis
Citation: Nasiru Salihu, Seyed Yaser Mousavi Siamakani, Auwal Bala Abubakar, Also Mohammed Saleh. A Hager–Zhang Riemannian conjugate gradient method for matrix approximation[J]. AIMS Mathematics, 2026, 11(5): 12580-12605. doi: 10.3934/math.2026517

Related Papers:

Abstract

We introduce a class of Hager–Zhang-type Riemannian conjugate gradient (CG) methods that generalize the framework of Sakai et al. (Applied Mathematics and Computation, 441 (2023) 127685) to arbitrary retractions while significantly extending its theoretical and practical scope. These methods ensure global convergence for non-convex problems without relying on strong convexity and inherently satisfy the sufficient descent property, independent of Riemannian line search conditions. A key algorithmic innovation is the introduction of an adaptive min(max) strategy to adjust the CG parameter $\beta_{k+1}$ using bounded parameters to ensure numerical stability. Furthermore, we extended classical Euclidean CG portfolio optimization to the sphere manifold, naturally enforcing budget constraints and improving robustness. Numerical experiments, available at GitHub repo, showed that our methods outperform classical Riemannian CG in iterations and computational time for large-scale problems.

References

[1]	P. A. Absil, R. Mahony, R. Sepulchre, Optimization algorithms on matrix manifolds, Princeton: Princeton University Press, 2009. https://doi.org/10.1515/9781400830244
[2]	N. Boumal, An introduction to optimization on smooth manifolds, Cambridge: Cambridge University Press, 2023. https://doi.org/10.1017/9781009166164
[3]	N. Andrei, Modern numerical nonlinear optimization, Cham: Springer, 2022. https://doi.org/10.1007/978-3-031-08720-2
[4]	W. Ring, B. Wirth, Optimization methods on Riemannian manifolds and their application to shape space, SIAM J. Optim., 22 (2012), 596–627. https://doi.org/10.1137/11082885X doi: 10.1137/11082885X
[5]	H. Sato, T. Iwai, A new, globally convergent Riemannian conjugate gradient method, Optimization, 64 (2015), 1011–1031. https://doi.org/10.1080/02331934.2013.836650 doi: 10.1080/02331934.2013.836650
[6]	Y. H. Dai, Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177–182. https://doi.org/10.1137/S1052623497318992 doi: 10.1137/S1052623497318992
[7]	H. Sakai, H. Iiduka, Hybrid Riemannian conjugate gradient methods with global convergence properties, Comput. Optim. Appl., 77 (2020), 811–830. https://doi.org/10.1007/s10589-020-00224-9 doi: 10.1007/s10589-020-00224-9
[8]	H. Sakai, H. Iiduka, Sufficient descent Riemannian conjugate gradient methods, J. Optim. Theory Appl., 190 (2021), 130–150. https://doi.org/10.1007/s10957-021-01874-3 doi: 10.1007/s10957-021-01874-3
[9]	C. Tang, W. Tan, S. Xing, H. Zheng, A class of spectral conjugate gradient methods for Riemannian optimization, Numer. Algor., 94 (2023), 131–147. https://doi.org/10.1007/s11075-022-01495-5 doi: 10.1007/s11075-022-01495-5
[10]	S. Nasiru, P. Kumam, S. Salisu, L. Wang, T. Seangwattana, Spectral conjugate gradient for Riemannian optimization: Application to the Gough–Stewart platform, Int. J. Dyn. Control, 13 (2025), 288. https://doi.org/10.1007/s40435-025-01788-2 doi: 10.1007/s40435-025-01788-2
[11]	N. Salihu, P. Kumam, S. M. Ibrahim, W. Kumam, Some combined techniques of spectral conjugate gradient methods with applications to robotic and image restoration models, Numer. Algor., 100 (2025), 553–593. https://doi.org/10.1007/s11075-024-01970-1 doi: 10.1007/s11075-024-01970-1
[12]	W. W. Hager, H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170–192. https://doi.org/10.1137/030601880 doi: 10.1137/030601880
[13]	W. W. Hager, H. Zhang, Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent, ACM Trans. Math. Software, 32 (2006), 113–137. https://doi.org/10.1145/1132973.1132979 doi: 10.1145/1132973.1132979
[14]	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
[15]	H. Sakai, H. Sato, H. Iiduka, Global convergence of Hager–Zhang type Riemannian conjugate gradient method, Appl. Math. Comput., 441 (2023), 127685. https://doi.org/10.1016/j.amc.2022.127685 doi: 10.1016/j.amc.2022.127685
[16]	N. Salihu, P. Kumam, S. Salisu, Two efficient nonlinear conjugate gradient methods for Riemannian manifolds, Comput. Appl. Math., 43 (2024), 415. https://doi.org/10.1007/s40314-024-02920-2 doi: 10.1007/s40314-024-02920-2
[17]	N. Salihu, P. Kumam, S. Salisu, L. Wang, K. Sitthithakerngkiet, A revised MRMIL Riemannian conjugate gradient method with simplified global convergence properties, Appl. Numer. Math., 214 (2025), 110–126. https://doi.org/10.1016/j.apnum.2025.03.007 doi: 10.1016/j.apnum.2025.03.007
[18]	N. Salihu, P. Kumam, I. M. Sulaiman, I. Arzuka, W. Kumam, An efficient Newton-like conjugate gradient method with restart strategy and its application, Math. Comput. Simulat., 226 (2024), 354–372. https://doi.org/10.1016/j.matcom.2024.07.008 doi: 10.1016/j.matcom.2024.07.008
[19]	N. Salihu, P. Kumam, L. Wang, M. M. Yahaya, An improved Dai–Kou conjugate gradient method with spectral search direction and applications, Eng. Optim., 58 (2026), 547–584. https://doi.org/10.1080/0305215X.2025.2475004 doi: 10.1080/0305215X.2025.2475004
[20]	X. Wang, A class of spectral three-term descent Hestenes–Stiefel conjugate gradient algorithms for large-scale unconstrained optimization and image restoration problems, Appl. Numer. Math., 192 (2023), 41–56. https://doi.org/10.1016/j.apnum.2023.05.024 doi: 10.1016/j.apnum.2023.05.024
[21]	A. M. Awwal, I. M. Sulaiman, M. Malik, M. Mamat, P. Kumam, K. Sitthithakerngkiet, A spectral RMIL+ conjugate gradient method for unconstrained optimization with applications in portfolio selection and motion control, IEEE Access, 9 (2021), 75398–75414. https://doi.org/10.1109/ACCESS.2021.3081570 doi: 10.1109/ACCESS.2021.3081570
[22]	M. Malik, I. M. Sulaiman, A. B. Abubakar, G. Ardaneswari, Sukono, A new family of hybrid three-term conjugate gradient method for unconstrained optimization with application to image restoration and portfolio selection, AIMS Mathematics, 8 (2023), 1–28. https://doi.org/10.3934/math.2023001 doi: 10.3934/math.2023001
[23]	N. Salihu, S. M. Ibrahim, P. Kaelo, I. A. Moghrabi, E. N. Madi, A spectral Fletcher–Reeves conjugate gradient method with integrated strategy for unconstrained optimization and portfolio selection, PloS One, 20 (2025), e0320416. https://doi.org/10.1371/journal.pone.0320416 doi: 10.1371/journal.pone.0320416
[24]	X. Zhu, A Riemannian conjugate gradient method for optimization on the Stiefel manifold, Comput. Optim. Appl., 67 (2017), 73–110. https://doi.org/10.1007/s10589-016-9883-4 doi: 10.1007/s10589-016-9883-4
[25]	B. Vandereycken, Low-rank matrix completion by Riemannian optimization, SIAM J. Optim., 23 (2013), 1214–1236. https://doi.org/10.1137/110845768 doi: 10.1137/110845768
[26]	H. Sakai, H. Iiduka, Riemannian adaptive optimization algorithm and its application to natural language processing, IEEE Trans. Cybern., 52 (2021), 7328–7339. https://doi.org/10.1109/TCYB.2021.3049845 doi: 10.1109/TCYB.2021.3049845
[27]	B. McCann, M. Nazari, Riemannian optimization techniques on Lie groups describing rigid body attitude and pose, J. Optim. Theory Appl., 207 (2025), 63. https://doi.org/10.1007/s10957-025-02810-5 doi: 10.1007/s10957-025-02810-5
[28]	H. Sato, Riemannian conjugate gradient methods: General framework and specific algorithms with convergence analyses, SIAM J. Optim., 32 (2022), 2690–2717. https://doi.org/10.1137/21M1464178 doi: 10.1137/21M1464178
[29]	J. K. Liu, S. J. Li, New hybrid conjugate gradient method for unconstrained optimization, Appl. Math. Comput., 245 (2014), 36–43. https://doi.org/10.1016/j.amc.2014.07.096 doi: 10.1016/j.amc.2014.07.096
[30]	J. Nocedal, S. J. Wright, Numerical optimization, 2 Eds., New York: Springer, 2006. https://doi.org/10.1007/978-0-387-40065-5
[31]	N. Boumal, B. Mishra, P. A. Absil, R. Sepulchre, Manopt: A MATLAB toolbox for optimization on manifolds, J. Mach. Learn. Res., 15 (2014), 1455–1459.
[32]	E. D. Dolan, J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201–213. https://doi.org/10.1007/s101070100263 doi: 10.1007/s101070100263
[33]	M. Bartholomew-Biggs, Nonlinear optimization with financial applications, New York: Springer, 2006. https://doi.org/10.1007/b102601
[34]	H. Markowitz, Portfolio selection, J. Finance, 7 (1952), 77–91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x doi: 10.1111/j.1540-6261.1952.tb01525.x
[35]	N. Salihu, P. Kumam, K. Sitthithakerngkiet, T. Seangwattana, A Riemannian conjugate gradient framework based on Dai–Liao principles for matrix approximation, IEEE Access, 13 (2025), 214813–214834. https://doi.org/10.1109/ACCESS.2025.3645695 doi: 10.1109/ACCESS.2025.3645695

Reader Comments

Your name:*

Email:*
© 2026 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)