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2011, 1(1): 71-82. doi: 10.3934/naco.2011.1.71

## A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations

 1 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China 2 College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

Received  October 2010 Revised  October 2010 Published  February 2011

In this paper, we propose a descent derivative-free method for solving symmetric nonlinear equations. The method is an extension of the modified Fletcher-Reeves (MFR) method proposed by Zhang, Zhou and Li [25] to symmetric nonlinear equations. It can be applied to solve large-scale symmetric nonlinear equations due to lower storage requirement. An attractive property of the method is that the directions generated by the method are descent for the residual function. By the use of some backtracking line search technique, the generated sequence of function values is decreasing. Under appropriate conditions, we show that the proposed method is globally convergent. The preliminary numerical results show that the method is practically effective.
Citation: Dong-Hui Li, Xiao-Lin Wang. A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 71-82. doi: 10.3934/naco.2011.1.71
##### References:
 [1] M. Al-Baali, Descent property and global convergence of the Fletcher-Reeves method with inexact line search,, IMA Journal of Numerical Analysis, 5 (1985), 121. doi: 10.1093/imanum/5.1.121. [2] S. Bellavia and B. Morini, A globally convergent Newton-GMRES subspace method for systems of nonlinear equations,, SIAM Journal on Scientific Computing, 23 (2001), 940. doi: 10.1137/S1064827599363976. [3] A. Griewank, The “global” convergence of Broyden-like methods with suitable line search,, Journal of Australia Mathematical Society, 28 (1986), 75. [4] W. Cheng and D. H. Li, A derivative-free nonmonotone line search and its application to the spectral residual method,, IMA Journal of Numerical Analysis, 29 (2009), 814. doi: 10.1093/imanum/drn019. [5] Y. H. Dai and Y. Yuan, Convergence of the Fletcher-Reeves method under a generalized Wolfe search,, Journal of Computational Mathematics, 2 (1996), 142. [6] Y. H. Dai and Y. Yuan, Convergence properties of the Fletcher-Reeves method,, IMA Journal of Numerical Analysis, 16 (1996), 155. doi: 10.1093/imanum/16.2.155. [7] Y. H. Dai and Y. Yuan, "Nonlinear Conjugate Gradient Methods,", Shanghai Science and Technology Publisher, (2000). [8] R. Fletcher and C. Reeves, Function minimization by conjugate gradients,, Computer Journal, 7 (1964), 149. doi: 10.1093/comjnl/7.2.149. [9] J. C. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization,, SIAM Journal on Optimization, 2 (1992), 21. doi: 10.1137/0802003. [10] G. Z. Gu, D. H. Li, L. Qi and S. Z. Zhou, Descent directions of Quasi-Newton methods for symmetric nonlinear equations,, SIAM Journal on Numerical Analysis, 40 (2003), 1763. doi: 10.1137/S0036142901397423. [11] J. Y. Han, G. H. Liu and H. X. Yin, Convergence properties of conjugate gradient methods with strong Wolfe linesearch,, Systems Science and Mathematical Science, 11 (1998), 112. [12] W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods,, Pacific Journal of Optimization, 2 (2006), 35. [13] Y. F. Hu and C. Storey, Global convergence result for conjugate gradient methods,, Journal of Optimization Theory and Applications, 71 (1991), 399. doi: 10.1007/BF00939927. [14] W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems,, Optimization Methods and Software, 18 (2003), 583. doi: 10.1080/10556780310001610493. [15] W. La Cruz, J.M. Martínez and M. Raydan, Spectral resdual method without gradient information for solving large-scale nonlinear systems of equations,, Mathematics of Computation, 75 (2006), 1429. doi: 10.1090/S0025-5718-06-01840-0. [16] G. H. Liu, J. Y. Han and H. X. Yin, Global convergence of the Fletcher-Reeves algorithm with an inexact line search,, Applied Mathematics, 10 (1995), 75. [17] D. H. Li and W. Cheng, Recent progress in the global convergence of quasi-Newton methods for nonlinear equations,, Hokkaido Journal of Mathematics, 36 (2007), 729. [18] D. H. Li and M. Fukushima, A globally and superlinearly convergent Gauss-Newton based BFGS method for symmetric nonlinear equations,, SIAM Journal on Numerical Analysis, 37 (1999), 152. doi: 10.1137/S0036142998335704. [19] D. H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like methods for nonlinear equations,, Optimization Methods and Software, 13 (2000), 181. doi: 10.1080/10556780008805782. [20] Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal of Numerical Analysis, (). [21] M. J. D. Powell, Some convergence properties of the conjugate gradient method,, Mathematical Programming, 11 (1976), 42. doi: 10.1007/BF01580369. [22] M. J . D. Powell, Restart procedures of the conjugate gradient method,, Mathematical Programming, 2 (1977), 241. doi: 10.1007/BF01593790. [23] Q. Yan, X. Z. Peng and D. H. Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations,, Journal of Computational and Applied Mathematics, 234 (2010), 649. doi: 10.1016/j.cam.2010.01.001. [24] J. Zhang and D. H. Li, A norm descent BFGS method for solving KKT systems of symmetric variational inequality problems,, Optimization Methods and Software, 22 (2007), 237. doi: 10.1080/10556780500397074. [25] L. Zhang, W. Zhou and D. H. Li, Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search,, Numerische Mathematik, 104 (2006), 561. doi: 10.1007/s00211-006-0028-z. [26] W. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations,, Journal of Computational Mathematics, 25 (2007), 89. [27] W. Zhou and D. H. Li, A globally convergent BFGS method for nonlinear monotone equations,, Mathematics of Computation, 77 (2008), 2231. doi: 10.1090/S0025-5718-08-02121-2. [28] G. Zoutendijk, Nonlinear Programming, Computational Methods,, in, (1970), 37.

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##### References:
 [1] M. Al-Baali, Descent property and global convergence of the Fletcher-Reeves method with inexact line search,, IMA Journal of Numerical Analysis, 5 (1985), 121. doi: 10.1093/imanum/5.1.121. [2] S. Bellavia and B. Morini, A globally convergent Newton-GMRES subspace method for systems of nonlinear equations,, SIAM Journal on Scientific Computing, 23 (2001), 940. doi: 10.1137/S1064827599363976. [3] A. Griewank, The “global” convergence of Broyden-like methods with suitable line search,, Journal of Australia Mathematical Society, 28 (1986), 75. [4] W. Cheng and D. H. Li, A derivative-free nonmonotone line search and its application to the spectral residual method,, IMA Journal of Numerical Analysis, 29 (2009), 814. doi: 10.1093/imanum/drn019. [5] Y. H. Dai and Y. Yuan, Convergence of the Fletcher-Reeves method under a generalized Wolfe search,, Journal of Computational Mathematics, 2 (1996), 142. [6] Y. H. Dai and Y. Yuan, Convergence properties of the Fletcher-Reeves method,, IMA Journal of Numerical Analysis, 16 (1996), 155. doi: 10.1093/imanum/16.2.155. [7] Y. H. Dai and Y. Yuan, "Nonlinear Conjugate Gradient Methods,", Shanghai Science and Technology Publisher, (2000). [8] R. Fletcher and C. Reeves, Function minimization by conjugate gradients,, Computer Journal, 7 (1964), 149. doi: 10.1093/comjnl/7.2.149. [9] J. C. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization,, SIAM Journal on Optimization, 2 (1992), 21. doi: 10.1137/0802003. [10] G. Z. Gu, D. H. Li, L. Qi and S. Z. Zhou, Descent directions of Quasi-Newton methods for symmetric nonlinear equations,, SIAM Journal on Numerical Analysis, 40 (2003), 1763. doi: 10.1137/S0036142901397423. [11] J. Y. Han, G. H. Liu and H. X. Yin, Convergence properties of conjugate gradient methods with strong Wolfe linesearch,, Systems Science and Mathematical Science, 11 (1998), 112. [12] W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods,, Pacific Journal of Optimization, 2 (2006), 35. [13] Y. F. Hu and C. Storey, Global convergence result for conjugate gradient methods,, Journal of Optimization Theory and Applications, 71 (1991), 399. doi: 10.1007/BF00939927. [14] W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems,, Optimization Methods and Software, 18 (2003), 583. doi: 10.1080/10556780310001610493. [15] W. La Cruz, J.M. Martínez and M. Raydan, Spectral resdual method without gradient information for solving large-scale nonlinear systems of equations,, Mathematics of Computation, 75 (2006), 1429. doi: 10.1090/S0025-5718-06-01840-0. [16] G. H. Liu, J. Y. Han and H. X. Yin, Global convergence of the Fletcher-Reeves algorithm with an inexact line search,, Applied Mathematics, 10 (1995), 75. [17] D. H. Li and W. Cheng, Recent progress in the global convergence of quasi-Newton methods for nonlinear equations,, Hokkaido Journal of Mathematics, 36 (2007), 729. [18] D. H. Li and M. Fukushima, A globally and superlinearly convergent Gauss-Newton based BFGS method for symmetric nonlinear equations,, SIAM Journal on Numerical Analysis, 37 (1999), 152. doi: 10.1137/S0036142998335704. [19] D. H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like methods for nonlinear equations,, Optimization Methods and Software, 13 (2000), 181. doi: 10.1080/10556780008805782. [20] Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal of Numerical Analysis, (). [21] M. J. D. Powell, Some convergence properties of the conjugate gradient method,, Mathematical Programming, 11 (1976), 42. doi: 10.1007/BF01580369. [22] M. J . D. Powell, Restart procedures of the conjugate gradient method,, Mathematical Programming, 2 (1977), 241. doi: 10.1007/BF01593790. [23] Q. Yan, X. Z. Peng and D. H. Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations,, Journal of Computational and Applied Mathematics, 234 (2010), 649. doi: 10.1016/j.cam.2010.01.001. [24] J. Zhang and D. H. Li, A norm descent BFGS method for solving KKT systems of symmetric variational inequality problems,, Optimization Methods and Software, 22 (2007), 237. doi: 10.1080/10556780500397074. [25] L. Zhang, W. Zhou and D. H. Li, Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search,, Numerische Mathematik, 104 (2006), 561. doi: 10.1007/s00211-006-0028-z. [26] W. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations,, Journal of Computational Mathematics, 25 (2007), 89. [27] W. Zhou and D. H. Li, A globally convergent BFGS method for nonlinear monotone equations,, Mathematics of Computation, 77 (2008), 2231. doi: 10.1090/S0025-5718-08-02121-2. [28] G. Zoutendijk, Nonlinear Programming, Computational Methods,, in, (1970), 37.
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