Some superlinearly convergent inexact generalized Newton method for solving nonsmooth equations

Abstract

This paper presents an inexact generalized Newton method for solving the nonlinear equation F(x)=0, where F is locally Lipschitz continuous. The method with backtracking is globally and superlinearly convergent under some mild assumptions on F. The first proposed algorithm is a substantial extension of the well-known inexact Newton method to nonsmooth case based on Pu and Tian [Globally convergent inexact generalized Newton's methods for nonsmooth equations, J. Comput. Appl. Math. 138 (2002), pp. 37–49] approach. Moreover, a hybrid method with Armijo line search, which is globally and quadratically convergent, is also presented. The presented results of numerical experiments are promising and confirm the theoretical properties of introduced methods.

Keywords:

• nonsmooth equation
• inexact generalized Newton method
• B-subdifferential
• superlinear convergence
• global convergence
• nonlinear complementarity problem

Mathematics Subject Classification :

• 65H10
• 90C53
• 90C33