An improved LS-RMIL-type conjugate gradient projection algorithm for systems of nonlinear equations and impulse noise image restoration

Yan Xia; Xuejie Ma; and Dandan Li; Yan Xia; Xuejie Ma; and Dandan Li

doi:10.3934/math.2025614

AIMS Mathematics

2025, Volume 10, Issue 6: 13640-13663. doi: 10.3934/math.2025614

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An improved LS-RMIL-type conjugate gradient projection algorithm for systems of nonlinear equations and impulse noise image restoration

School of Artificial Intelligence, Guangzhou Huashang College, Guangzhou 511300, China

Received: 18 April 2025 Revised: 30 May 2025 Accepted: 09 June 2025 Published: 13 June 2025
MSC : 65K05, 90C56

This paper proposes an improved LS-RMIL-type conjugate gradient projection algorithm designed for solving systems of nonlinear equations with convex constraints. The algorithm introduces a search direction that maintains sufficient descent and trust-region properties independent of the line search approach. It operates under relatively mild conditions, requiring only continuity and monotonicity of nonlinear equations, thus avoiding the need for stronger assumptions such as Lipschitz continuity. The global convergence of the algorithm is established under these relaxed conditions. Furthermore, numerical experiments demonstrate that the algorithm exhibits superior efficiency and stability, particularly in solving large-scale nonlinear systems and in applications such as impulse noise image restoration, outperforming existing methods.
- conjugate gradient projection method,
- systems of nonlinear equations,
- global convergence,
- impulse noise image restoration
Citation: Yan Xia, Xuejie Ma, and Dandan Li. An improved LS-RMIL-type conjugate gradient projection algorithm for systems of nonlinear equations and impulse noise image restoration[J]. AIMS Mathematics, 2025, 10(6): 13640-13663. doi: 10.3934/math.2025614

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Abstract

This paper proposes an improved LS-RMIL-type conjugate gradient projection algorithm designed for solving systems of nonlinear equations with convex constraints. The algorithm introduces a search direction that maintains sufficient descent and trust-region properties independent of the line search approach. It operates under relatively mild conditions, requiring only continuity and monotonicity of nonlinear equations, thus avoiding the need for stronger assumptions such as Lipschitz continuity. The global convergence of the algorithm is established under these relaxed conditions. Furthermore, numerical experiments demonstrate that the algorithm exhibits superior efficiency and stability, particularly in solving large-scale nonlinear systems and in applications such as impulse noise image restoration, outperforming existing methods.

References

[1]	H. Chen, Y. Wang, H. Zhao, Finite convergence of a projected proximal point algorithm for the generalized variational inequalities, Oper. Res. Lett., 40 (2012), 303–305. https://doi.org/10.1016/j.orl.2012.03.011 doi: 10.1016/j.orl.2012.03.011
[2]	Y. Wang, L. Qi, S. Luo, An alternative steepest direction method for the optimization in evaluating geometric discord, Pacific J. Optim., 10 (2014), 137–149.
[3]	N. Iusem, V. Solodov, Newton-type methods with generalized distances for constrained optimization, Optimization, 41 (1997), 257–278. https://doi.org/10.1080/02331939708844339 doi: 10.1080/02331939708844339
[4]	P. Liu, L. Li, H. Shao, M. Liu, J. Fan, An inertial-type CG projection method with restart for pseudo-monotone costs with application to traffic assignment, Netw. Spat. Econ., 25 (2024), 147–172. https://doi.org/10.1007/s11067-024-09653-z doi: 10.1007/s11067-024-09653-z
[5]	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
[6]	Y. Zhao, D. Li, Monotonicity of fixed point and normal mappings associated with variational inequality and its application, SIAM J. Optim., 11 (2001), 962–973. https://doi.org/10.1137/S1052623499357957 doi: 10.1137/S1052623499357957
[7]	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
[8]	D. Li, Y. Li, S. Wang, An improved three-term conjugate gradient algorithm for constrained nonlinear equations under non-lipschitz conditions and its Applications, Mathematics, 12 (2024), 2556. https://doi.org/10.3390/math12162556 doi: 10.3390/math12162556
[9]	Z. Liu, X. Zhang, M. Su, Convergence analysis of Newton-Raphson method in feasible power-flow for DC network, IEEE Trans. Power Syst., 35 (2020), 4100–4103. https://doi.org/10.1109/TPWRS.2020.2986706 doi: 10.1109/TPWRS.2020.2986706
[10]	N. Barnafi, L. Pavarino, S. Scacchi, Parallel inexact Newton-Krylov and quasi-Newton solvers for nonlinear elasticity, Comput. Methods Appl. Mech. Eng., 400 (2022) 115557. https://doi.org/10.1016/j.cma.2022.115557 doi: 10.1016/j.cma.2022.115557
[11]	A. Mahdavi, S. Salehi, A superlinearly convergent nonmonotone quasi-Newton method for unconstrained multiobjective optimization, Optim. Methods Software, 35 (2020) 1223–1247. https://doi.org/10.1080/10556788.2020.1737691 doi: 10.1080/10556788.2020.1737691
[12]	R. Sihwail, O. Solaiman, K. Omar, A hybrid approach for solving systems of nonlinear equations using harris hawks optimization and newtons method, IEEE Access, 9 (2021), 95791–95807. https://doi.org/10.1109/ACCESS.2021.3094471 doi: 10.1109/ACCESS.2021.3094471
[13]	V. Krutikov, E. Tovbis, P. Stanimirovic, On the convergence rate of quasi-Newton methods on strongly convex functions with lipschitz gradient, Mathematics, 11 (2023), 4715. https://doi.org/10.3390/math11234715 doi: 10.3390/math11234715
[14]	G. Yuan, T. Li, W. Hu, A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems, Appl. Numer. Math., 147 (2020), 129–141. https://doi.org/10.1016/j.apnum.2019.08.022 doi: 10.1016/j.apnum.2019.08.022
[15]	M. Koorapetse, P. Kaelo, An efficient hybrid conjugate gradient-based projection method for convex constrained nonlinear monotone equations, J. Interdiscip. Math., 22 (2019), 1031–1050. https://doi.org/10.1080/09720502.2019.1700889 doi: 10.1080/09720502.2019.1700889
[16]	K. Danmalam, H. Mohammad, A. Abubakar, Hybrid algorithm for system of nonlinear monotone equations based on the convex combination of Fletcher-Reeves and a new conjugate residual parameters, Thai J. Math., 18 (2020), 2093–2106.
[17]	M. Waziri, A. Yusuf, A. Abubakar, Improved conjugate gradient method for nonlinear system of equations, Comput. Appl. Math., 39 (2020), 321. https://doi.org/10.1007/s40314-020-01374-6 doi: 10.1007/s40314-020-01374-6
[18]	T. Song, Z. Liu, An efficient subspace minimization conjugate gradient method for solving nonlinear monotone equations with convex constraints, Axioms, 13 (2024), 170. https://doi.org/10.3390/axioms13030170 doi: 10.3390/axioms13030170
[19]	G. Ma, J. Jin, 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
[20]	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
[21]	J. Sabiu, A. Shah, P. Stanimirovic, Modified optimal Perry conjugate gradient method for solving system of monotone equations with applications, Appl. Numer. Math., 184 (2023), 431–445. https://doi.org/10.1016/j.apnum.2022.10.016 doi: 10.1016/j.apnum.2022.10.016
[22]	Y. Liu, C. Storey, Efficient generalized conjugate gradient algorithms, part 1: Theory, J. Optim. Theory Appl., 69 (1991), 129–137. https://doi.org/10.1007/BF00940464 doi: 10.1007/BF00940464
[23]	M. Rivaie, M. Mamat, L. June, 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]	P. Liu, H. Shao, Z. Yuan, X. Wu, 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
[25]	P. Liu, X. Wu, H. Shao, Y. Zhang, S. Cao, Three adaptive hybrid derivative-free projection methods for constrained monotone nonlinear equations and their applications, Numer. Linear Algebra Appl., 30 (2023), e2471. https://doi.org/10.1002/nla.2471 doi: 10.1002/nla.2471
[26]	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
[27]	A. Ibrahim, P. Kumam, W. Kumam, A family of derivative-free conjugate gradient methods for constrained nonlinear equations and image restoration, IEEE Access, 8 (2020), 162714–162729. https://doi.org/10.1109/ACCESS.2020.3020969 doi: 10.1109/ACCESS.2020.3020969
[28]	E. 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
[29]	R. Chan, C. Ho, M. Nikolova, Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization, IEEE Trans. Image Process., 14 (2005), 1479–1485. https://doi.org/10.1109/TIP.2005.852196 doi: 10.1109/TIP.2005.852196
[30]	J. Cai, R. Chan, D. Fiore, Minimization of a detail-preserving regularization functional for impulse noise removal, J. Math. Imaging Vis., 29 (2007), 79–91. https://doi.org/10.1007/s10851-007-0027-4 doi: 10.1007/s10851-007-0027-4

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