A BFGS algorithm for solving symmetric nonlinear equations

In this article, we propose a BFGS method for solving symmetric nonlinear equations. The presented method possesses some favourable properties: (a) the generated sequence of iterates is norm descent; (b) the generated sequence of the quasi-Newton matrix is positive definite and (c) this method possesses the global convergence and superlinear convergence. Numerical results show that the presented method is interesting.

[1]  Zengxin Wei,et al.  New line search methods for unconstrained optimization , 2009 .

[2]  Masao Fukushima,et al.  A Globally and Superlinearly Convergent Gauss-Newton-Based BFGS Method for Symmetric Nonlinear Equations , 1999, SIAM J. Numer. Anal..

[3]  Detong Zhu,et al.  Nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations , 2005, Appl. Math. Comput..

[4]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[5]  Anderas Griewank.  The “global” convergence of Broyden-like methods with suitable line search , 1986, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[6]  Gonglin Yuan,et al.  A Rank-One Fitting Method with Descent Direction for Solving Symmetric Nonlinear Equations , 2009, Int. J. Commun. Netw. Syst. Sci..

[7]  Gonglin Yuan,et al.  A New Method with Descent Property for Symmetric Nonlinear Equations , 2010 .

[8]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[9]  Zengxin Wei,et al.  The superlinear convergence analysis of a nonmonotone BFGS algorithm on convex objective functions , 2008 .

[10]  Xiwen Lu,et al.  A new backtracking inexact BFGS method for symmetric nonlinear equations , 2008, Comput. Math. Appl..

[11]  Yong Wang,et al.  A new trust region method for nonlinear equations , 2003, Math. Methods Oper. Res..

[12]  J. J. Moré,et al.  Quasi-Newton Methods, Motivation and Theory , 1974 .

[13]  Yu-Hong Dai,et al.  Convergence Properties of the BFGS Algoritm , 2002, SIAM J. Optim..

[14]  Jorge J. Moré,et al.  Benchmarking optimization software with performance profiles , 2001, Math. Program..

[15]  Dong-Hui Li,et al.  Descent Directions of Quasi-Newton Methods for Symmetric Nonlinear Equations , 2002, SIAM J. Numer. Anal..

[16]  J. Nocedal,et al.  A tool for the analysis of Quasi-Newton methods with application to unconstrained minimization , 1989 .

[17]  Zengxin Wei,et al.  MODIFIED LIMITED MEMORY BFGS METHOD WITH NONMONOTONE LINE SEARCH FOR UNCONSTRAINED OPTIMIZATION , 2010 .

[18]  Stephen J. Wright,et al.  Quasi-Newton Methods , 1999 .

[19]  Xiwen Lu,et al.  A conjugate gradient method with descent direction for unconstrained optimization , 2009, J. Comput. Appl. Math..

[20]  Roger Fletcher,et al.  Practical methods of optimization; (2nd ed.) , 1987 .

[21]  Mengnien Wu,et al.  Numerical Schubert Calculus by the Pieri Homotopy Algorithm , 2002, SIAM J. Numer. Anal..

[22]  Yousef Saad,et al.  Convergence Theory of Nonlinear Newton-Krylov Algorithms , 1994, SIAM J. Optim..

[23]  Jin-yanFan.  A MODIFIED LEVENBERG-MARQUARDT ALGORITHM FOR SINGULAR SYSTEM OF NONLINEAR EQUATIONS , 2003 .

[24]  Masao Fukushima,et al.  Testing Parallel Variable Transformation , 1999, Comput. Optim. Appl..

[25]  Gonglin Yuan,et al.  Convergence analysis of a modified BFGS method on convex minimizations , 2010, Comput. Optim. Appl..

[26]  Gonglin Yuan,et al.  A Trust-Region-Based BFGS Method with Line Search Technique for Symmetric Nonlinear Equations , 2009, Adv. Oper. Res..

[27]  Gonglin Yuan,et al.  Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems , 2009, Optim. Lett..

[28]  H. Walker.  Quasi-Newton Methods , 1978 .

[29]  Xiwen Lu,et al.  A modified PRP conjugate gradient method , 2009, Ann. Oper. Res..

[30]  J. J. Moré,et al.  A Characterization of Superlinear Convergence and its Application to Quasi-Newton Methods , 1973 .

[31]  Xi-wen,et al.  A Nonmonotone Trust Region Method for Solving Symmetric Nonlinear Equations , 2009 .

[32]  Xiwen Lu,et al.  BFGS trust-region method for symmetric nonlinear equations , 2009 .

[33]  Jinyan Fan,et al.  An improved trust region algorithm for nonlinear equations , 2011, Comput. Optim. Appl..