A convergence analysis of hybrid gradient projection algorithm for constrained nonlinear equations with applications in compressed sensing. (English) Zbl 1533.65084
Summary: In this paper, we propose a projection-based hybrid spectral gradient algorithm for nonlinear equations with convex constraints, which is based on a certain line search strategy. Convex combination technique is used to design a novel spectral parameter that is inspired by some classical spectral gradient methods. The search direction also meets the sufficient descent condition and trust region feature. The global convergence of the proposed algorithm has been established under reasonable assumptions. The results of the experiment demonstrate the proposed algorithm is both more promising and robust than some similar methods, and it is also capable of handling large-scale optimization problems. Furthermore, we apply it to problems involving sparse signal recovery and blurred image restoration.
MSC:
| 65K05 | Numerical mathematical programming methods |
| 65H10 | Numerical computation of solutions to systems of equations |
| 90C30 | Nonlinear programming |
| 90C56 | Derivative-free methods and methods using generalized derivatives |
Keywords:
constrained nonlinear equations; spectral gradient method; global convergence; convex combination technique; compressed sensingSoftware:
MCPLIBReferences:
| [1] | Iusem, NA; Solodov, VM, Newton-type methods with generalized distances for constrained optimization, Optimization, 41, 3, 257-278 (1997) · Zbl 0905.49015 · doi:10.1080/02331939708844339 |
| [2] | Ghaddar, B.; Marecek, J.; Mevissen, M., Optimal power flow as a polynomial optimization problem, IEEE Trans. Power Syst., 31, 1, 539-546 (2015) · doi:10.1109/TPWRS.2015.2390037 |
| [3] | Dirkse, SP; Ferris, MC, MCPLIB: a collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5, 4, 319-345 (1995) · doi:10.1080/10556789508805619 |
| [4] | Zhao, YB; Li, D., Monotonicity of fixed point and normal mappings associated with variational inequality and its application, SIAM J. Optim., 11, 4, 962-973 (2001) · Zbl 1010.90084 · doi:10.1137/S1052623499357957 |
| [5] | Zheng, L.; Yang, L.; Liang, Y., A modified spectral gradient projection method for solving non-linear monotone equations with convex constraints and its application, IEEE Access, 8, 92677-92686 (2020) |
| [6] | Yin, J.h. Jian, J.b. Jiang, X.z. A generalized hybrid CGPM-based algorithm for solving large-scale convex constrained equations with applications to image restoration. J Comput Appl Math 391, 113423 (2021) · Zbl 1464.65069 |
| [7] | Yin, J.; Jian, J.; Jiang, X.; Liu, M.; Wang, L., A hybrid three-term conjugate gradient projection method for constrained nonlinear monotone equations with applications, Numer. Algorithms, 88, 1, 389-418 (2021) · Zbl 1484.65131 · doi:10.1007/s11075-020-01043-z |
| [8] | Abubakar, AB; Kumam, P.; Ibrahim, AH; Rilwan, J., Derivative-free HS-DY-type method for solving nonlinear equations and image restoration, Heliyon, 6, 11, e05400 (2020) · doi:10.1016/j.heliyon.2020.e05400 |
| [9] | Hu, WJ; Wu, JZ; Yuan, GL, Some modified Hestenes-Stiefel conjugate gradient algorithms with application in image restoration, Appl. Numer. Math., 158, 360-376 (2020) · Zbl 1450.90028 · doi:10.1016/j.apnum.2020.08.009 |
| [10] | Yang, L.; Chen, Y.; Tong, X.; Deng, C., A new smoothing Newton method for solving constrained nonlinear equations, Appl. Math. Comput., 217, 24, 9855-9863 (2011) · Zbl 1261.65051 |
| [11] | Ling, C.; Yin, H.; Zhou, G., A smoothing Newton-type method for solving the \(l_2\) spectral estimation problem with lower and upper bounds, Comput. Optim. Appl., 50, 2, 351-378 (2011) · Zbl 1261.90056 · doi:10.1007/s10589-010-9356-0 |
| [12] | Qi, L.; Tong, XJ; Li, DH, Active-set projected trust-region algorithm for box-constrained nonsmooth equations, J. Optim. Theory Appl., 120, 3, 601-625 (2004) · Zbl 1140.65331 · doi:10.1023/B:JOTA.0000025712.43243.eb |
| [13] | Kimiaei, M., A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints, Calcolo, 54, 3, 769-812 (2017) · Zbl 1373.90151 · doi:10.1007/s10092-016-0208-x |
| [14] | Yu, Z. Lin, J. Sun, J. Xiao, Y. Liu, L, Li, Z. Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59(10), 2416-2423 (2009) · Zbl 1183.65056 |
| [15] | Xiao, Y.; Zhu, H., A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405, 1, 310-319 (2013) · Zbl 1316.90050 · doi:10.1016/j.jmaa.2013.04.017 |
| [16] | Sun, M.; Liu, J., Three derivative-free projection methods for nonlinear equations with convex constraints, J. Appl. Math. Comput., 47, 1, 265-276 (2015) · Zbl 1348.90630 · doi:10.1007/s12190-014-0774-5 |
| [17] | Barzilai, J.; Borwein, JM, Two-point step size gradient methods, IMA J. Numer. Anal., 8, 1, 141-148 (1988) · Zbl 0638.65055 · doi:10.1093/imanum/8.1.141 |
| [18] | Raydan, M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., 7, 1, 26-33 (1997) · Zbl 0898.90119 · doi:10.1137/S1052623494266365 |
| [19] | Narushima, Y.; Wakamatsu, T.; Yabe, H., Extended Barzilai-Borwein method for unconstrained minimization problems, Pacific J. Optim., 6, 3, 591-613 (2008) · Zbl 1225.90080 |
| [20] | Cruz, WL; Raydan, M., Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Methods Softw., 18, 5, 583-599 (2003) · Zbl 1069.65056 · doi:10.1080/10556780310001610493 |
| [21] | Li, X.; Guo, X., Spectral residual methods with two new non-monotone line searches for large-scale nonlinear systems of equations, Appl. Math. Comput., 269, 59-69 (2015) · Zbl 1410.90217 |
| [22] | Dai, YH; Huang, Y.; Liu, XW, A family of spectral gradient methods for optimization, Comput. Optim. Appl., 74, 1, 43-65 (2019) · Zbl 1427.90260 · doi:10.1007/s10589-019-00107-8 |
| [23] | Yu, Z.; Li, L.; Li, P., A family of modified spectral projection methods for nonlinear monotone equations with convex constraint, Math. Probl. Eng., 2020, 3, 1-16 (2020) · Zbl 1459.65089 |
| [24] | Abubakar, AB; Kumam, P.; Mohammad, H., A note on the spectral gradient projection method for nonlinear monotone equations with applications, Comput. Appl. Math., 39, 2, 1-35 (2020) · Zbl 1449.65116 · doi:10.1007/s40314-020-01151-5 |
| [25] | Mohammad, H.; Abubakar, AB, A positive spectral gradient-like method for large-scale nonlinear monotone equations, Bull. Comput. Appl. Math., 5, 1, 99-115 (2017) · Zbl 1398.65105 |
| [26] | Abubakar, AB; Kumam, P.; Mohammad, H.; Awwal, AM, A Barzilai-Borwein gradient projection method for sparse signal and blurred image restoration, J. Franklin Inst., 357, 11, 7266-7285 (2020) · Zbl 1455.94004 · doi:10.1016/j.jfranklin.2020.04.022 |
| [27] | Liu, J.; Duan, Y., Two spectral gradient projection methods for constrained equations and their linear convergence rate, J. Inequal. Appl., 2015, 1, 1-13 (2015) · Zbl 1310.65066 · doi:10.1186/s13660-014-0525-z |
| [28] | Kumam, P.; Abubakar, AB; Ibrahim, AH; Kura, HU; Panyanak, B.; Pakkaranang, N., Another hybrid approach for solving monotone operator equations and application to signal processing, Math. Methods Appl. Sci., 45, 12, 789-7922 (2022) · Zbl 1527.90223 · doi:10.1002/mma.8285 |
| [29] | Ibrahim, A.; Kuman, P.; Abubakar, A.; Abubakar, J., A derivative-free projection method for nonlinear equations with non-Lipschitz operator: application to LASSO problem (2023), Sci: Math. Methods Appl, Sci · Zbl 1527.65036 |
| [30] | Ibrahim, AH; Deepho, J.; Abubakar, AB; Kamandi, A., A globally convergent derivative-free projection algorithm for signal processing, J. Interdiscip. Math., 25, 8, 2301-2320 (2022) · doi:10.1080/09720502.2021.1960000 |
| [31] | Ibrahim, A.H. Kumam, P. Abubakar, A.B. Abdullahi, M.S. Mohammad, H. A Dai-Liao-type projection method for monotone nonlinear equations and signal processing. Demonstr. Math. 2022, 55(1), 978-1013 (2022) · Zbl 1510.90261 |
| [32] | Yuan, G.; Li, T.; Hu, W., A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems, Appl. Numer. Math., 147, 129-141 (2020) · Zbl 1433.90165 · doi:10.1016/j.apnum.2019.08.022 |
| [33] | Abubakar, AB; Kumam, P.; Awwal, AM, Global convergence via descent modified three-term conjugate gradient projection algorithm with applications to signal recovery, Results Appl. Math., 4, 100069 (2019) · Zbl 1453.65120 · doi:10.1016/j.rinam.2019.100069 |
| [34] | Gao, P.; He, C., An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints, Calcolo, 55, 4, 1-17 (2018) · Zbl 1405.65043 · doi:10.1007/s10092-018-0291-2 |
| [35] | Liu, J.; Feng, Y., A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numer. Algorithms, 82, 1, 245-262 (2019) · Zbl 1431.65073 · doi:10.1007/s11075-018-0603-2 |
| [36] | Dolan, ED; Moré, JJ, Benchmarking optimization software with performance profiles, Math. Program., 91, 2, 201-213 (2002) · Zbl 1049.90004 · doi:10.1007/s101070100263 |
| [37] | Donoho, DL, Compressed sensing, IEEE Trans. Inf. Theory, 52, 4, 1289-1306 (2006) · Zbl 1288.94016 · doi:10.1109/TIT.2006.871582 |
| [38] | Solodov, MV; Svaiter, BF, A new projection method for variational inequality problems, SIAM J. Control Optim., 37, 3, 765-776 (1999) · Zbl 0959.49007 · doi:10.1137/S0363012997317475 |
| [39] | Xiao, Y.; Wang, Q.; Hu, Q., Non-smooth equations based method for \(l_1\)-norm problems with applications to compressed sensing, Nonlinear Anal., 74, 11, 3570-3577 (2011) · Zbl 1217.65069 · doi:10.1016/j.na.2011.02.040 |
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