Partially symmetrical derivative-free Liu-Storey projection method for convex constrained equations. (English) Zbl 1499.90221
Summary: Applying Powell symmetrical technique to the Liu-Storey conjugate gradient method, a partially symmetrical Liu-Storey conjugate gradient method is proposed and extended to solve nonlinear monotone equations with convex constraints, which satisfies the sufficient descent condition without any line search. By using some line searches, the global convergence is proved merely by assuming that the equations are Lipschitz continuous. Moreover, we prove the R-linear convergence rate of the proposed method with an additional assumption. Finally, compared with one existing method, the performance of the proposed method is showed by some numerical experiments on the given test problems.
MSC:
| 90C30 | Nonlinear programming |
| 65K05 | Numerical mathematical programming methods |
| 65H10 | Numerical computation of solutions to systems of equations |
Keywords:
nonlinear monotone equations; conjugate gradient method; symmetrical technique; projection technique; global convergenceSoftware:
minpackReferences:
| [1] | Cheng, W. Y.; Li, D. H., An active set modified PolakCRibiéreCPolyak method for large-scale nonlinear bound constrained optimization, J. Optim. Theory Appl., 155, 1084-1084 (2012) · Zbl 1276.90067 · doi:10.1007/s10957-012-0091-9 |
| [2] | Dai, Z. F.; Chen, X. H.; Wen, F. H., A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations, Appl. Math. Comput., 270, 378-386 (2015) · Zbl 1410.90248 |
| [3] | Gabriel, S.A. and Pang, J.S., A trust region method for constrained nonsmooth equations, in Large Scale Optimization-State of the Art, W.W. Hager, D.W. Hearn, P.M. Pardalos, eds., Kluwer, Dordrecht, 1994, pp. 155-181. · Zbl 0813.65091 |
| [4] | Kanzow, C.; Yamashita, N.; Fukushima, M., Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties, J. Comput. Appl. Math., 172, 375-397 (2004) · Zbl 1064.65037 · doi:10.1016/j.cam.2004.02.013 |
| [5] | La Cruz, W., A projected derivative-free algorithm for nonlinear equations with convex constraints, Optim. Methods Softw., 29, 583-599 (2014) · Zbl 1285.65028 · doi:10.1080/10556788.2012.721129 |
| [6] | La Cruz, W.; Martínez, J. M.; Raydan, M., Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comput., 75, 1429-1448 (2006) · Zbl 1122.65049 · doi:10.1090/S0025-5718-06-01840-0 |
| [7] | Liu, J. K., Derivative-free spectral PRP projection method for solving nonlinear monotone equations, Math.Numer. Sinica (Chinese), 2, 113-124 (2016) · Zbl 1363.65090 |
| [8] | Liu, J. K.; Li, S. J., A projection method for convex constrained monotone nonlinear equations with applications, Comput. Math. Appl., 70, 2442-2453 (2015) · Zbl 1443.65073 · doi:10.1016/j.camwa.2015.09.014 |
| [9] | Liu, J. K.; Li, S. J., Multivative spectral DY-type projection method for nonlinear monotone equations, J. Ind. Manag. Optim., 13, 283-296 (2017) · Zbl 1368.49038 |
| [10] | Liu, Y.; Storey, C., Efficient generalized conjugate gradient algorithms, J. Optim. Theory. Appl., 69, 129-137 (1992) · Zbl 0702.90077 · doi:10.1007/BF00940464 |
| [11] | Moŕe, J. J.; Garbow, B. S.; Hillström, K. E., Testing unconstrained optimization software, ACM Trans. Math. Softw., 7, 17-41 (1981) · Zbl 0454.65049 · doi:10.1145/355934.355936 |
| [12] | Powell, M.J.D., A new algorithm for unconstrained optimization, in Nonlinear Programming, J.B. Rosen, O.L. Mangasarian, K. Ritter, eds., Academic Press, New York, 1970, pp. 31-66. · Zbl 0228.90043 |
| [13] | Qi, L.; Tong, X. J.; Li, D. H., An active-set projected trust region algorithm for box constrained nonsmooth equations, J. Optim. Theory Appl., 120, 601-625 (2004) · Zbl 1140.65331 · doi:10.1023/B:JOTA.0000025712.43243.eb |
| [14] | Solodov, M.V. and Svaiter, B.F., A globally convergent inexact newton method for system of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth, and Smoothing Methods, M. Fukushima and L. Qi, eds., Kluwer Academic Publishers, Dordrecht, Holland, 1999, pp. 355-369. · Zbl 0928.65059 |
| [15] | Tong, X. J.; Qi, L.; Yang, Y. F., The Lagrangian globalization method for nonsmooth constrained equations, Comput. Optim. Appl., 33, 89-109 (2006) · Zbl 1103.90077 · doi:10.1007/s10589-005-5960-9 |
| [16] | Tong, X. J.; Zhou, S. Z., A smoothing projected Newton-type method for semismooth equations with bound constraints, J. Ind. Manag. Optim., 1, 235-250 (2005) · Zbl 1177.90389 · doi:10.3934/jimo.2005.1.235 |
| [17] | Xiao, Y. H.; Zhu, H., A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405, 310-319 (2013) · Zbl 1316.90050 · doi:10.1016/j.jmaa.2013.04.017 |
| [18] | Yu, Z. S.; Lin, J.; Sun, J.; XIao, Y. X.; Liu, L. Y.; Li, Z. H., Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59, 2416-2423 (2009) · Zbl 1183.65056 · doi:10.1016/j.apnum.2009.04.004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
