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

Gonglin YUAN, Zhongxing WANG, Zengxin WEI

.

DOI: 10.4236/ijcns.2009.26061                

Abstract

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.

Keywords

Rank-One Update, Global Convergence, Nonlinear Equations, Descent Direction

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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.