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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

Dong-Hui Li 1, and  Xiao-Lin Wang 2,

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.
Keywords: Symmetric nonlinear equations, descent direction, derivative-free method, global convergence..
Mathematics Subject Classification: Primary: 65H10; Secondary: 90C3.
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), 121-124. doi: 10.1093/imanum/5.1.121.  Google Scholar

[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), 940-960. doi: 10.1137/S1064827599363976.  Google Scholar

[3]

A. Griewank, The “global” convergence of Broyden-like methods with suitable line search, Journal of Australia Mathematical Society, Ser. B, 28 (1986), 75-92.  Google Scholar

[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), 814-825. doi: 10.1093/imanum/drn019.  Google Scholar

[5]

Y. H. Dai and Y. Yuan, Convergence of the Fletcher-Reeves method under a generalized Wolfe search, Journal of Computational Mathematics, 2 (1996), 142-148.  Google Scholar

[6]

Y. H. Dai and Y. Yuan, Convergence properties of the Fletcher-Reeves method, IMA Journal of Numerical Analysis, 16 (1996), 155-164. doi: 10.1093/imanum/16.2.155.  Google Scholar

[7]

Y. H. Dai and Y. Yuan, "Nonlinear Conjugate Gradient Methods," Shanghai Science and Technology Publisher, Shanghai, 2000. Google Scholar

[8]

R. Fletcher and C. Reeves, Function minimization by conjugate gradients, Computer Journal, 7 (1964), 149-154. doi: 10.1093/comjnl/7.2.149.  Google Scholar

[9]

J. C. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM Journal on Optimization, 2 (1992), 21-42. doi: 10.1137/0802003.  Google Scholar

[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), 1763-1774. doi: 10.1137/S0036142901397423.  Google Scholar

[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), 112-116.  Google Scholar

[12]

W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.  Google Scholar

[13]

Y. F. Hu and C. Storey, Global convergence result for conjugate gradient methods, Journal of Optimization Theory and Applications, 71 (1991), 399-405. doi: 10.1007/BF00939927.  Google Scholar

[14]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods and Software, 18 (2003), 583-599. doi: 10.1080/10556780310001610493.  Google Scholar

[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), 1429-1448. doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[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, Journal of Chinese Universities, Ser. B, 10 (1995), 75-82.  Google Scholar

[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), 729-743.  Google Scholar

[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), 152-172. doi: 10.1137/S0036142998335704.  Google Scholar

[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), 181-201. doi: 10.1080/10556780008805782.  Google Scholar

[20]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal of Numerical Analysis, ().   Google Scholar

[21]

M. J. D. Powell, Some convergence properties of the conjugate gradient method, Mathematical Programming, 11 (1976), 42-49. doi: 10.1007/BF01580369.  Google Scholar

[22]

M. J . D. Powell, Restart procedures of the conjugate gradient method, Mathematical Programming, 2 (1977), 241-254. doi: 10.1007/BF01593790.  Google Scholar

[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), 649-657. doi: 10.1016/j.cam.2010.01.001.  Google Scholar

[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), 237-252. doi: 10.1080/10556780500397074.  Google Scholar

[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), 561-572. doi: 10.1007/s00211-006-0028-z.  Google Scholar

[26]

W. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations, Journal of Computational Mathematics, 25 (2007), 89-96.  Google Scholar

[27]

W. Zhou and D. H. Li, A globally convergent BFGS method for nonlinear monotone equations, Mathematics of Computation, 77 (2008), 2231-2240. doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

[28]

G. Zoutendijk, Nonlinear Programming, Computational Methods, in "Integer and Nonlinear Programming" (eds. J. Abadie), North-Holland, Amsterdam, (1970), 37-86.  Google Scholar

show all references

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), 121-124. doi: 10.1093/imanum/5.1.121.  Google Scholar

[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), 940-960. doi: 10.1137/S1064827599363976.  Google Scholar

[3]

A. Griewank, The “global” convergence of Broyden-like methods with suitable line search, Journal of Australia Mathematical Society, Ser. B, 28 (1986), 75-92.  Google Scholar

[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), 814-825. doi: 10.1093/imanum/drn019.  Google Scholar

[5]

Y. H. Dai and Y. Yuan, Convergence of the Fletcher-Reeves method under a generalized Wolfe search, Journal of Computational Mathematics, 2 (1996), 142-148.  Google Scholar

[6]

Y. H. Dai and Y. Yuan, Convergence properties of the Fletcher-Reeves method, IMA Journal of Numerical Analysis, 16 (1996), 155-164. doi: 10.1093/imanum/16.2.155.  Google Scholar

[7]

Y. H. Dai and Y. Yuan, "Nonlinear Conjugate Gradient Methods," Shanghai Science and Technology Publisher, Shanghai, 2000. Google Scholar

[8]

R. Fletcher and C. Reeves, Function minimization by conjugate gradients, Computer Journal, 7 (1964), 149-154. doi: 10.1093/comjnl/7.2.149.  Google Scholar

[9]

J. C. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM Journal on Optimization, 2 (1992), 21-42. doi: 10.1137/0802003.  Google Scholar

[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), 1763-1774. doi: 10.1137/S0036142901397423.  Google Scholar

[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), 112-116.  Google Scholar

[12]

W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.  Google Scholar

[13]

Y. F. Hu and C. Storey, Global convergence result for conjugate gradient methods, Journal of Optimization Theory and Applications, 71 (1991), 399-405. doi: 10.1007/BF00939927.  Google Scholar

[14]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods and Software, 18 (2003), 583-599. doi: 10.1080/10556780310001610493.  Google Scholar

[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), 1429-1448. doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[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, Journal of Chinese Universities, Ser. B, 10 (1995), 75-82.  Google Scholar

[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), 729-743.  Google Scholar

[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), 152-172. doi: 10.1137/S0036142998335704.  Google Scholar

[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), 181-201. doi: 10.1080/10556780008805782.  Google Scholar

[20]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal of Numerical Analysis, ().   Google Scholar

[21]

M. J. D. Powell, Some convergence properties of the conjugate gradient method, Mathematical Programming, 11 (1976), 42-49. doi: 10.1007/BF01580369.  Google Scholar

[22]

M. J . D. Powell, Restart procedures of the conjugate gradient method, Mathematical Programming, 2 (1977), 241-254. doi: 10.1007/BF01593790.  Google Scholar

[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), 649-657. doi: 10.1016/j.cam.2010.01.001.  Google Scholar

[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), 237-252. doi: 10.1080/10556780500397074.  Google Scholar

[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), 561-572. doi: 10.1007/s00211-006-0028-z.  Google Scholar

[26]

W. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations, Journal of Computational Mathematics, 25 (2007), 89-96.  Google Scholar

[27]

W. Zhou and D. H. Li, A globally convergent BFGS method for nonlinear monotone equations, Mathematics of Computation, 77 (2008), 2231-2240. doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

[28]

G. Zoutendijk, Nonlinear Programming, Computational Methods, in "Integer and Nonlinear Programming" (eds. J. Abadie), North-Holland, Amsterdam, (1970), 37-86.  Google Scholar

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