A new backtracking inexact BFGS method for symmetric nonlinear equations

A BFGS method, in association with a new backtracking line search technique, is presented for solving symmetric nonlinear equations. The global and superlinear convergences of the given method are established under mild conditions. Preliminary numerical results show that the proposed method is better than the normal technique for the given problems.

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

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

[3]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

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

[5]  M. Powell.  A New Algorithm for Unconstrained Optimization , 1970 .

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

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

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

[9]  Gonglin Yuan,et al.  The Superlinear Convergence of a Modified BFGS-Type Method for Unconstrained Optimization , 2004, Comput. Optim. Appl..

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

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

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

[13]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

[14]  M. Fukushima,et al.  A modified BFGS method and its global convergence in nonconvex minimization , 2001 .

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

[16]  Liqun Qi,et al.  An SQP-Type Method and Its Application in Stochastic Programs , 2003 .

[17]  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.

[18]  P. Toint,et al.  Local convergence analysis for partitioned quasi-Newton updates , 1982 .

[19]  J. Nocedal,et al.  Global Convergence of a Class of Quasi-newton Methods on Convex Problems, Siam Some Global Convergence Properties of a Variable Metric Algorithm for Minimization without Exact Line Searches, Nonlinear Programming, Edited , 1996 .

[20]  Guoyin Li,et al.  New quasi-Newton methods for unconstrained optimization problems , 2006, Appl. Math. Comput..

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