Gonglin Yuan, Laisheng Yu

Abstract: In this paper, we propose a new method which based on the nonmonotone line search technique for solving symmetric nonlinear equations. The method can ensure that the search direction is descent for the norm function. Under suitable conditions, the global convergence of the method is proved. Numerical results show that the presented method is practicable for the test problems.

Keywords: Nonmonotone Line Search, Symmetric Equations, Global Convergence

Cite this paper: nullG. Yuan and L. Yu, "A Nonmonotone Line Search Method for Symmetric Nonlinear Equations," Intelligent Control and Automation, Vol. 1 No. 1, 2010, pp. 28-35. doi: 10.4236/ica.2010.11004.

References

[1]   D. Li and M. Fukushima, “A Global And Superlinear Convergent Gauss-Newton-based BFGS Method for Symmetric Nonlinear Equations,” SIAM Journal on Numerical Analysis, Vol. 37, No. 1, 1999, pp. 152-172.

[2]   Z. Wei, G. Yuan and Z. Lian, “An Approximate Gauss- Newton-Based BFGS Method for Solving Symmetric Nonlinear Equations,” Guangxi Sciences, Vol. 11, No. 2, 2004, pp. 91-99.

[3]   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, 2004, pp. 10-26.

[4]   P. N. Brown and Y. Saad, “Convergence Theory of Nonlinear Newton-Krylov Algorithms,” SIAM Journal on Optimization, Vol. 4, 1994, pp. 297-330.

[5]   D. Zhu, “Nonmonotone Backtracking Inexact Quasi-Newton Algorithms for Solving Smooth Nonlinear Equations,” Applied Mathematics and Computation, Vol. 161, No. 3, 2005, pp. 875-895.

[6]   G. Yuan and X. Lu, “A New Backtracking Inexact BFGS Method for Symmetric Nonlinear Equations,” Computers and Mathematics with Applications, Vol. 55, No. 1, 2008, pp. 116-129.

[7]   S. G. Nash, “A Survey of Truncated-Newton Methods,” Journal of Computational and Applied Mathematics, Vol. 124, No. 1-2, 2000, pp. 45-59.

[8]   A. Griewank, “The ‘Global’ Convergence of Broyden- Like Methods with a Suitable Line Search,” Journal of the Australian Mathematical Society Series B, Vol. 28, No. 1, 1986, pp. 75-92.

[9]   A. R. Conn, N. I. M. Gould and P. L. Toint, “Trust Region Method,” Society for Industrial and Applied Mathe- matics, Philadelphia, 2000.

[10]   J. E. Dennis and R. B. Schnabel, “Numerical Methods for Unconstrained Optimization and Nonlinear Equations,” Englewood Cliffs, Prentice-Hall, 1983.

[11]   K. Levenberg, “A Method for The Solution of Certain Nonlinear Problem in Least Squares,” Quarterly of Applied Mathematics, Vol. 2, 1944, pp. 164-166.

[12]   D. W. Marquardt, An algorithm for least-squares estimation of nonlinear inequalities, SIAM Journal on Applied Mathematics, Vol. 11, 1963, pp. 431-441.

[13]   J. Nocedal and S. J. Wright, “Numerical Optimization,” Springer, New York, 1999.

[14]   N. Yamashita and M. Fukushima, “On the Rate of Convergence of the Levenberg-Marquardt Method,” Computing, Vol. 15, No. Suppl, 2001, pp. 239-249.

[15]   Y. Yuan and W. Sun, “Optimization Theory and Algorithm,” Scientific Publisher House, Beijing, 1997.

[16]   G. Yuan and X. Li, “A Rank-One Fitting Method for Solving Symmetric Nonlinear Equations,” Journal of Applied Functional Analysis, Vol. 5, No. 4, 2010, pp. 389- 407.

[17]   G. Yuan, X. Lu and Z. Wei, “BFGS Trust-Region Method for Symmetric Nonlinear Equations,” Journal of Computational and Applied Mathematics, Vol. 230, No. 1, 2009, pp. 44-58.

[18]   G. Yuan, S. Meng and Z. Wei, “A Trust-Region-Based BFGS Method with Line Search Technique for Symmetric Nonlinear Equations,” Advances in Operations Research, Vol. 2009, 2009, pp. 1-20.

[19]   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.

[20]   G. Yuan, Z. Wei and X. Lu, “A Nonmonotone Trust Region Method for Solving Symmetric Nonlinear Equations,” Chinese Quarterly Journal of Mathematics, Vol. 24, No. 4, 2009, pp. 574-584.

[21]   H. Zhang and W. Hager, “A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization,” SIAM Journal on Optimization, Vol. 14, No. 4, 2004, pp. 1043-1056.

[22]   J. M. Ortega and W. C. Rheinboldt, “Iterative Solution of Nonlinear Equations in Several Variables,” Academic Press, New York, 1970.

[23]   E. Yamakawa and M. Fukushima, “Testing Parallel Bariable Transformation,” Computational Optimization and Applications, Vol. 13, 1999, pp. 253-274.