A Hybrid Line Search And Global Convergence Of Quasi-Newton Method For Nonlinear Equations

Quasi-newton method is a kind of effective methods for solving nonlinear equations. Linear search is an important way to ensure the global convergence of quasi-newton method. On the one hand, the monotone linear search techniques lies in that the generated sequence of function values is monotonically decreasing. However, a monotone linear search may generate a small step-length and need more trial steps. The non-monotone linear search techniques can decrease trial steps and generate a bigger step-length. However, a non-monotone linear search techniques can't generate a monotonically decreasing sequence of function values. Because the quasi-newton direction for solving nonlinear equations is not usually decreasing direction, so there are still no well-done monotone linear search.On the other hand,non-monotone linear search of quasi-newton method for solving non-linear equations has achieved important results. Broyden's rank one method with different non-monotone linear search may converges globally and superlinearly.In this paper, we further study the linear search technique for solving nonlin-ear equations. We propose a hybrid linear search technique for nonlinear equations, and the linear search have an important nature:When the quasi-newton direction is decreasing and a bigger step is available, we use monotone line search. Other-wise, we use non-monotonic linear search. Because the quasi-newton method for solving nonlinear equations is a derivative-free method, so we can judge descenting property of quasi-newton direction under the condition of excluding derivative of model function.Through combining the linear search and finite difference, we propose a deriva-tive free criterion which can judge whether the quasi-newton direction is decreasing. In appropriate conditions, we show that the Broyden's rank one method with this hybrid linear search converges globally and superlinearly. Finally, we use numerical experiments to test the proposed hybrid linear search, the results of numerical experiments show that the Broyden's rank one method with hybrid linear search is superior to the proposed monotone linear search quasi-newton method and non-monotone linear search quasi-newton method.