A matrix-free DFP-like optimization method for problems arising from compressive sensing

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper introduces a matrix-free variant of the Davidon-Fletcher-Powell (DFP) method for unconstrained optimization problems with applications in compressive sensing and image restoration. The main contribution lies in the new search direction incorporating a scaling parameter that ensures the satisfaction of the sufficient descent condition, independent of the line search conditions. A rigorous convergence analysis guarantees the boundedness and theoretical validity of the proposed method. Comprehensive numerical experiments on benchmark unconstrained optimization test problems and compressive sensing problems demonstrate the efficiency and robustness of the algorithm. Specifically, in image restoration tasks, our method outperforms CG-DESCENT, MDL, and NSMA, achieving a 100% success rate compared to 95.8%, 84.5%, and 53.5%, respectively. Additionally, results on computational time, relative error, and PSNR confirm the superior performance of the proposed approach. These findings establish the proposed method as a competitive alternative for large-scale optimization problems.

Keywords

three-term conjugate gradient algorithms / optimization models / global convergence / line search procedure / signal processing

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] P. Singh, S. S. Bose, A quantum-clustering optimization method for COVID-19 CT scan image segmentation, Expert Syst. Appl., 185 (2021), 115637. https://doi.org/10.1016/j.eswa.2021.115637

[2] P. Singh, M. K. Muchahari, Solving multi-objective optimization problem of convolutional neural network using fast forward quantum optimization algorithm: Application in digital image classification, Adv. Eng. Software, 176 (2023), 103370. https://doi.org/10.1016/j.advengsoft.2022.103370

[3] 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. Simul., 226 (2024), 354-372. https://doi.org/10.1016/j.matcom.2024.07.008

[4] M. Malik, S. S. Abas, M. Mamat, I. S. Mohammed, A new hybrid conjugate gradient method with global convergence properties, Int. J. Adv. Sci. Technol., 29 (2020), 199-210.

[5] W. Sun, Y. Yuan, Optimization Theory and Methods: Nonlinear Programming , Springer, 2006.

[6] A. M. Awwal, P. Kumam, K. Sitthithakerngkiet, A. M. Bakoji, A. S. Halilu, I. M. Sulaiman, Derivative-free method based on dfp updating formula for solving convex constrained nonlinear monotone equations and application, AIMS Math., 6 (2021), 8792-8814. https://doi.org/10.3934/math.2021510

[7] L. Zhang, W. Zhou, D. Li, A descent modified polak-ribiére-polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal., 26 (2006), 629-640. https://doi.org/10.1093/imanum/drl016

[8] A. M. Awwal, L. Wang, P. Kumam, M. I. Sulaiman, S. Salisu, N. Salihu, et al., Generalized RMIL conjugate gradient method under the strong wolfe line search with application in image processing, Math. Methods Appl. Sci., 46 (2023), 17544-17556. https://doi.org/10.1002/mma.9515

[9] S. M. Ibrahim, N. Salihu, Two sufficient descent spectral conjugate gradient algorithms for unconstrained optimization with application, Optim. Eng., 26 (2025), 655-679. https://doi.org/10.1007/s11081-024-09899-z

[10] A. U. Omesa, S. M. Ibrahim, R. B. Yunus, I. A. Moghrab, M. Y. Waziri, A. Sambas, A brief survey of line search methods for optimization problems, Results Control Optim., 19 (2025), 100550. https://doi.org/10.1016/j.rico.2025.100550

[11] N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl., 38 (2007), 401-416. https://doi.org/10.1007/s10589-007-9055-7

[12] P. Wolfe, Convergence conditions for ascent methods, SIAM Rev., 11 (1969), 226-235. https://doi.org/10.1137/1011036

[13] P. Wolfe, Convergence conditions for ascent methods. ii: Some corrections, SIAM Rev., 13 (1971), 185-188. https://doi.org/10.1137/1013035

[14] G. Zoutendijk, Nonlinear programming, computational methods, Integer Nonlinear Program., 1970 (1970), 37-86.

[15] 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

[16] J. Zhang, Y. Xiao, Z. Wei, Nonlinear conjugate gradient methods with sufficient descent condition for large-scale unconstrained optimization, Math. Probl. Eng., 2009 (2009), 243290. https://doi.org/10.1155/2009/243290

[17] M. Jourak, S. Nezhadhosein, F. Rahpeymaii, A new self-scaling memoryless quasi-newton update for unconstrained optimization, 4OR, 22 (2024), 235-252. https://doi.org/10.1007/s10288-023-00544-6

[18] N. Andrei, An unconstrained optimization test functions collection, Adv. Model. Optim., 10 (2008), 147-161.

[19] E. D. Dolan, J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213. https://doi.org/10.1007/s101070100263

[20] B. Sun, H. Feng, K. Chen, X. Zhu, A deep learning framework of quantized compressed sensing for wireless neural recording, IEEE Access, 4 (2016), 5169-5178. https://doi.org/10.1109/ACCESS.2016.2604397

[21] I. F. Akyildiz, M. C. Vuran, Wireless Sensor Networks , John Wiley & Sons, 2010.

[22] Y. J. J. Bagul, A smooth transcendental approximation to |x|, Int. J. Math. Sci. Eng. Appls. (IJMSEA), 11 (2017), 213-217.

[23] A. M. Awwal, M. M. Yahaya, N. Pakkaranang, N. Pholasa, A new variant of the conjugate descent method for solving unconstrained optimization problems and applications, Mathematics (2227-7390), 12 (2024), 2430. https://doi.org/10.3390/math12152430

[24] Z. Aminifard, S. Babaie-Kafaki, A nonmonotone admm-based diagonal quasi-newton update with application to the compressive sensing problem, Math. Model. Anal., 28 (2023), 673-688. https://doi.org/10.3846/mma.2023.16993

[25] Z. Aminifard, A. Hosseini, S. Babaie-Kafaki, Modified conjugate gradient method for solving sparse recovery problem with nonconvex penalty, Signal Process., 193 (2022), 108424. https://doi.org/10.1016/j.sigpro.2021.108424

[26] 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., 2024 (2024), 1-41. https://doi.org/10.1007/s11075-024-01970-1

[27] Y. Liu, Z. Zhu, B. Zhang, Two sufficient descent three-term conjugate gradient methods for unconstrained optimization problems with applications in compressive sensing, J. Appl. Math. Comput., 68 (2022), 1787-1816. https://doi.org/10.1007/s12190-021-01589-8

[28] X. Jiang, W. Liao, J. Yin, J. Jian, A new family of hybrid three-term conjugate gradient methods with applications in image restoration, Numer. Algor., 91 (2022), 161-191. https://doi.org/10.1007/s11075-022-01258-2