Scientific Research

An Academic Publisher

A Rank-One Fitting Method with Descent Direction for Solving Symmetric Nonlinear Equations

**Author(s)**Leave a comment

In this paper, a rank-one updated method for solving symmetric nonlinear equations is proposed. This method possesses some features: 1) The updated matrix is positive definite whatever line search technique is used; 2) The search direction is descent for the norm function; 3) The global convergence of the given method is established under reasonable conditions. Numerical results show that the presented method is interesting.

Cite this paper

G. YUAN, Z. WANG and Z. WEI, "A Rank-One Fitting Method with Descent Direction for Solving Symmetric Nonlinear Equations,"

*International Journal of Communications, Network and System Sciences*, Vol. 2 No. 6, 2009, pp. 555-561. doi: 10.4236/ijcns.2009.26061.

[1] | R. Fletcher, Practical meethods of optimization, 2nd Edition, John Wiley & Sons, Chichester, 1987. |

[2] | A. Griewank and L. Toint, “Local convergence analysis for partitioned quasi-Newton updates,” Numerical Mathematics, No. 39, pp. 429–448, 1982. |

[3] | G. L. Yuan and X. W. Lu, “A new line search method with trust region for unconstrained optimization,” Communications on Applied Nonlinear Analysis, Vol. 15, No. 1, pp. 35–49, 2008. |

[4] | G. L. Yuan and X. W. Lu, “A modified PRP conjugate gradient method,” Annals of Operations Research, No. 166, pp. 73–90, 2009. |

[5] | G. L. Yuan, X. W. Lu, and Z. X. Wei, “New two-point step size gradient methods for solving unconstrained optimization problems,” Natural Science Journal of Xiang-tan University, Vol. 1, No. 29, pp. 13–15, 2007. |

[6] | G. L. Yuan, X. W. Lu, and Z. X. Wei, “A conjugate gradient method with descent direction for unconstrained optimization,” Journal of Computational and Applied Mathematics, No. 233, pp. 519–530, 2009. |

[7] | G. L. Yuan and Z. X. Wei, “New line search methods for unconstrained optimization,” Journal of the Korean Statistical Society, No. 38, pp. 29–39, 2009. |

[8] | G. L. Yuan and Z. X. Wei, “A rank-one fitting method for unconstrained optimization problems,” Mathematica Applicata, Vol. 1, No. 22, pp. 118–122, 2009. |

[9] | G. L. Yuan and Z. X. Wei, “A nonmonotone line search method for regression analysis,” Journal of Service Science and Management, Vol. 1, No. 2, pp. 36–42, 2009. |

[10] | R. Byrd and J. Nocedal, “A tool for the analysis of quasi-Newton methods with application to unconstrained minimization,” SIAM Journal on Numerical Analysis, No. 26, pp. 727–739, 1989. |

[11] | R. Byrd, J. Nocedal, and Y. Yuan, “Global convergence of a class of quasi-Newton methods on convex problems,” SIAM Journal on Numerical Analysis, No. 24, pp. 1171–1189, 1987. |

[12] | Y. Dai, “Convergence properties of the BFGS algorithm,” SIAM Journal on Optimization, No. 13, pp. 693– 701, 2003. |

[13] | J. E. Dennis and J. J. More, “A characterization of super-linear convergence and its application to quasi-Newtion methods,” Mathematics of Computation, No. 28, pp. 549–560, 1974. |

[14] | J. E. Dennis and R. B. Schnabel, “Numerical methods for unconstrained optimization and nonlinear equations,” Pretice-Hall, Inc., Englewood Cliffs, NJ, 1983. |

[15] | M. J. D. Powell, “A new algorithm for unconstrained optimation,” in Nonlinear Programming, J. B. Rosen, O. L. Mangasarian and K. Ritter, eds. Academic Press, New York, 1970. |

[16] | Y. Yuan and W. Sun, Theory and Methods of Optimization, Science Press of China, 1999. |

[17] | G. L. Yuan and Z. X. Wei, “The superlinear convergence analysis of a nonmonotone BFGS algorithm on convex,” Objective Functions, Acta Mathematica Sinica, English Series, Vol. 24, No. 1, pp. 35–42, 2008. |

[18] | D. Li and M. Fukushima, “A modified BFGS method and its global convergence in nonconvex minimization,” Journal of Computational and Applied Mathematics, No. 129, pp. 15–35, 2001. |

[19] | D. Li and M. Fukushima, “On the global convergence of the BFGS methods for on convex unconstrained optimization problems,” SIAM Journal on Optimization, No. 11, pp. 1054–1064, 2001. |

[20] | Z. Wei, G. Li, and L. Qi, “New quasi-Newton methods for unconstrained optimization problems,” Applied Mathematics and Computation, No. 175, pp. 1156–1188, 2006. |

[21] | Z. Wei, G. Yu, G. Yuan, and Z. Lian, “The superlinear convergence of a modified BFGS-type method for un-constrained optimization,” Computational Optimization and Applications, No. 29, pp. 315–332, 2004. |

[22] | G. L. Yuan and Z. X. Wei, “Convergence analysis of a modified BFGS method on convex minimizations,” Computational Optimization and Applications, doi: 10.1007/ s10 589–008–9219–0. |

[23] | J. Z. Zhang, N. Y. Deng, and L. H. Chen, “New quasi- Newton equation and related methods for unconstrained optimization,” Journal of Optimization Theory and Ap-plications, No. 102, pp. 147–167, 1999. |

[24] | Y. Xu and C. Liu, “A rank-one fitting algorithm for unconstrained optimization problems,” Applied Mathematics and Letters, No. 17, pp. 1061–1067, 2004. |

[25] | A. Griewank, “The ‘global’ convergence of Broyden-like methods with a suitable line search,” Journal of the Australian Mathematical Society, Series B., No. 28, pp. 75– 92, 1986. |

[26] | Y. Yuan, “Trust region algorithm for nonlinear equations, information,” No. 1, pp. 7–21, 1998. |

[27] | G. L. Yuan, X. W. Lu, and Z. X. Wei, “BFGS trust-region method for symmetric nonlinear equations,” Journal of Computational and Applied Mathematics, No. 230, pp. 44–58, 2009. |

[28] | J. Zhang and Y. Wang, “A new trust region method for nonlinear equations,” Mathematical Methods of Opera-tions Research, No. 58, pp. 283–298, 2003. |

[29] | D. Zhu, “Nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations,” Applied Mathematics and Computation, No. 161, pp. 875– 895, 2005. |

[30] | D. Li and M. Fukushima, “A global and superlinear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations,” SIAM Journal on Numerical Analysis, No. 37, pp. 152–172, 1999. |

[31] | G. Yuan and X. Li, “An approximate Gauss-Newton- based BFGS method with descent directions for solving symmetric nonlinear equations,” OR Transactions, Vol. 8, No. 4, pp. 10–26, 2004. |

[32] | G. L. Yuan and X. W. Lu, “A new backtracking inexact BFGS method for symmetric nonlinear equations,” Com-puter and Mathematics with Application, No. 55, pp. 116–129, 2008. |

[33] | P. N. Brown and Y. Saad, “Convergence theory of nonlinear Newton-Kryloy algorithms,” SIAM Journal on Optimization, No. 4, pp. 297–330, 1994. |

[34] | G. Gu, D. Li, L. Qi, and S. Zhou, “Descent directions of quasi-Newton methods for symmetric nonlinear equa-tions,” SIAM Journal on Numerical Analysis, Vol. 5, No. 40, pp. 1763–1774, 2002. |

[35] | G. Yuan, “Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems,” Optimization Letters, No. 3, pp. 11– 21, 2009. |

[36] | J. J. More, B. S. Garow, and K. E. Hillstrome, “Testing unconstrained optimization software,” ACM Transactions on Mathematical Software, No. 7, pp. 17–41, 1981. |

Copyright © 2017 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.