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A NEW DERIVATIVE-FREE CONJUGATE GRADIENT METHOD FOR LARGE-SCALE NONLINEAR SYSTEMS OF EQUATIONS

Published online by Cambridge University Press:  22 March 2017

XIAOWEI FANG  and
QIN NI
XIAOWEI FANG*
Affiliation:
College of Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Department of Mathematics, Huzhou University, Huzhou 313000, China email fangxiaowei@163.com
QIN NI
Affiliation:
College of Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China email niqfs@nuaa.edu.cn
*
✉fangxiaowei@163.com
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Abstract

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We propose a new derivative-free conjugate gradient method for large-scale nonlinear systems of equations. The method combines the Rivaie–Mustafa–Ismail–Leong conjugate gradient method for unconstrained optimisation problems and a new nonmonotone line-search method. The global convergence of the proposed method is established under some mild assumptions. Numerical results using 104 test problems from the CUTEst test problem library show that the proposed method is promising.

Keywords

large-scale nonlinear systems of equationsconjugate gradient methodnonmonotone line search

MSC classification

Primary: 90C56: Derivative-free methods and methods using generalized derivatives
Secondary: 90C30: Nonlinear programming
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 3 , June 2017 , pp. 500 - 511
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported by the National Natural Science Foundation of China (11071117, 11274109), the Natural Science Foundation of Jiangsu Province (BK20141409), and the Natural Science Foundation of Huzhou University (KX21072).

References

Al-Baali, M., Spedicato, E. and Maggioni, F., ‘Broyden’s quasi-Newton methods for a nonlinear system of equations and unconstrained optimization: a review and open problems’, Optim. Methods Softw. 29 (2014), 937–954.CrossRefGoogle Scholar
Cheng, W., ‘A two-term PRP-based descent method’, Numer. Funct. Anal. Optim. 28 (2007), 1217–1230.CrossRefGoogle Scholar
Cheng, W. and Li, D. H., ‘A derivative-free nonmonotone line search and its application to the spectral residual method’, IMA J. Numer. Anal. 29 (2009), 814–825.CrossRefGoogle Scholar
Cheng, W., Xiao, Y. and Hu, Q. J., ‘A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations’, J. Comput. Appl. Math. 224 (2009), 11–19.CrossRefGoogle Scholar
Cruz, W. L., Martínez, J. and Raydan, M., ‘Spectral residual method without gradient information for solving large-scale nonlinear systems of equations’, Math. Comp. 75 (2006), 1429–1448.CrossRefGoogle Scholar
Cruz, W. L. and Raydan, M., ‘Nonmonotone spectral methods for large-scale nonlinear systems’, Optim. Methods Softw. 18 (2003), 583–599.CrossRefGoogle Scholar
Dolan, E. D. and Moré, J. J., ‘Benchmarking optimization software with performance profiles’, Math. Program. 91 (2001), 201–213.CrossRefGoogle Scholar
Gould, N. I. M., Orban, D. and Toint, Ph. L., ‘CUTEst: a constrained and unconstrained testing environment with safe threads for mathematical optimization’, Comput. Optim. Appl. 60 (2015), 545–557.CrossRefGoogle Scholar
Li, G., Tang, C. M. and Wei, Z. X., ‘New conjugacy condition and related new conjugate gradient methods for unconstrained optimization’, J. Comput. Appl. Math. 202(2) (2007), 523–539.CrossRefGoogle Scholar
Min, L., ‘A derivative-free PRP method for solving large-scale nonlinear systems of equations and its global convergence’, Optim. Methods Softw. 29 (2014), 503–514.Google Scholar
Polak, E., Optimization: Algorithms and Consistent Approximations (Springer, New York, 2012).Google Scholar
Rivaie, M., Mamat, M., June, L. W. and Mohd, I., ‘A new class of nonlinear conjugate gradient coefficients with global convergence properties’, Appl. Math. Comput. 218 (2012), 11323–11332.Google Scholar
Wei, Z. X., Li, G. and Qi, L. Q., ‘Global convergence of the Polak–Ribière–Polyak conjugate gradient method with inexact line searches for nonconvex unconstrained optimization problems’, Math. Comp. 77 (2008), 2173–2193.CrossRefGoogle Scholar
Yu, G., Guan, L. and Li, G., ‘Global convergence of modified Polak–Ribière–Polyak conjugate gradient methods with sufficient descent property’, J. Ind. Manag. Optim. 4(3) (2008), 565–579.CrossRefGoogle Scholar
Zhang, H. and Hager, W. W., ‘A nonmonotone line search technique and its application to unconstrained optimization’, SIAM J. Optim. 14 (2004), 1043–1056.CrossRefGoogle Scholar
Zhang, L., Zhou, W. and Li, D. H., ‘A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence’, IMA J. Numer. Anal. 26 (2006), 629–640.CrossRefGoogle Scholar