Limited memory BFGS method with backtracking for symmetric nonlinear equations. (English) Zbl 1225.65056
Summary: A limited memory BFGS algorithm is presented for solving symmetric nonlinear equations. The global convergence of the given method is established under mild conditions. Numerical results show that the limited memory BFGS method is more interesting than the normal BFGS method for the given problems.
MSC:
| 65H10 | Numerical computation of solutions to systems of equations |
References:
| [1] | Ortega, J. M.; Rheinboldt, W. C., Iterative of Nonlinear Equations in Several Variables (1970), Academic Press: Academic Press New York · Zbl 0241.65046 |
| [2] | Fan, J. Y., A modified Levenberg-Marquardt algorithm for singular system of nonlinear equations, Journal of Computer Mathematics, 21, 625-636 (2003) · Zbl 1032.65053 |
| [3] | Yuan, Y., Trust region algorithm for nonlinear equations, Information, 1, 7-21 (1998) |
| [4] | Zhang, J.; Wang, Y., A new trust region method for nonlinear equations, Mathematical Methods of Operations Research, 58, 283-298 (2003) · Zbl 1043.65072 |
| [5] | Brown, P. N.; Saad, Y., Convergence theorey of nonlinear Newton-Kryloy algorithms, SIAM Journal on Optimization, 4, 297-330 (1994) · Zbl 0814.65048 |
| [6] | Zhu, D., Nonmonotone backtracking inexact quasi-Newton algorithms for solving smmoth nonlinear equations, Applied Mathematics and Computation, 161, 875-895 (2005) · Zbl 1073.65047 |
| [7] | Yuan, G.; Lu, X., A new backtracking inexact BFGS method for symmetric nonlinear equations, Computers and Mathematics with Applications, 55, 116-129 (2008) · Zbl 1176.65063 |
| [8] | Li, D.; Fukushima, M., A global and superlinear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations, SIAM Journal on Numerical Analysis, 37, 152-172 (1999) · Zbl 0946.65031 |
| [9] | Yuan, G.; Li, X., A rank-one fitting method for solving symmetric nonlinear equations, Journal of Applied Functional Analysis, 5, 389-407 (2010) · Zbl 1223.90048 |
| [10] | Li, D.; Qi, L.; Zhou, S., Descent directions of quasi-Newton methods for symmetric nonlinear equations, SIAM Journal on Numerical Analysis, 40, 1763-1774 (2002) · Zbl 1047.65032 |
| [11] | Yuan, G., A new method with descent property for symmetric nonlinear equations, Numerical Functional Analysis and Optimization, 31, 974-987 (2010) · Zbl 1201.90164 |
| [12] | Yuan, G.; Lu, X.; Wei, Z., BFGS trust-region method for symmetric nonlinear equations, Journal of Computational and Applied Mathematics, 230, 44-58 (2009) · Zbl 1168.65026 |
| [13] | Nash, S. G., A surey of truncated-Newton matrices, Journal of Computational and Applied Mathematics, 124, 45-59 (2000) · Zbl 0969.65054 |
| [14] | Griewank, A., The ‘global’ convergence of Broyden-like methods with a suitable line search, Journal of the Australian Mathematical Society, Series B, 28, 75-92 (1986) · Zbl 0596.65034 |
| [15] | Byrd, R. H.; Nocedal, J.; Schnabel, R. B., Representations of quasi-Newton matrices and their use in limited memory methods, Mathematical Programming, 63, 129-156 (1994) · Zbl 0809.90116 |
| [16] | Xiao, Y.; Li, D., An active set limited memory BFGS algorithm for large-scale bound constrained optimization, Mathematical Methods of Operations Research, 67, 443-454 (2008) · Zbl 1145.90084 |
| [17] | Xiao, Y.; Wei, Z., A new subspace limited memory BFGS algorithm for large-scale bound constrained optimization, Applied Mathematics and Computation, 185, 350-359 (2007) · Zbl 1114.65069 |
| [18] | Fletcher, R., Practical Meethods of Optimization (1987), John Wiley and Sons: John Wiley and Sons Chichester · Zbl 0905.65002 |
| [19] | Byrd, R.; Nocedal, J., A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM Journal on Numerical Analysis, 26, 727-739 (1989) · Zbl 0676.65061 |
| [20] | Yamakawa, E.; Fukushima, M., Testing parallel bariable transformation, Computational Optimization and Applications, 13, 253-274 (1999) · Zbl 1040.90545 |
| [21] | Gomez-Ruggiero, M.; Martinez, J. M.; Moretti, A., Comparing algorithms for solving sparse nonlinear systems of equations, SIAM Journal on Scienitific and Statistical Computing, 23, 459-483 (1992) · Zbl 0752.65039 |
| [22] | Raydan, M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal on Optimization, 7, 26-33 (1997) · Zbl 0898.90119 |
| [23] | Li, G., Successive column correction algorithms for solving sparse nonlinear systems of equations, Mathematical Programming, 43, 187-207 (1989) · Zbl 0675.65045 |
| [24] | Bing, Y.; Lin, G., An efficient implementation of Merrill’s method for sparse or partially separable systems of nonlinear equations, SIAM Journal on Optimization, 2, 206-221 (1991) · Zbl 0754.65048 |
| [25] | Moré, J. J.; Garbow, B. S.; Hillström, K. E., Testing uncosntrained optimization software, ACM Transactions on Mathematical Software, 7, 17-41 (1981) · Zbl 0454.65049 |
| [26] | Roberts, S. M.; Shipman, J. J., On the closed form solution of Troeschs problem, Journal of Computational Physics, 21, 291-304 (1976) · Zbl 0334.65062 |
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