An Inexact Regularized Newton Method For Monotone Nonlinear Equations

Study on numerical algorithms for solving nonlinear equations is an important research direction of computational mathematics.This paper discusses a class of special nonlinear equations which possesses monotone structure.These problems have important research background,for examples,the first order necessary condition of convex optimization problem,some monotone variational inequality problems can be converted into monotone nonlinear equations equivalently.Newton method is an efficient algorithm for solving nonlinear equations,which often requires computing a linear equations exactly each iteration.This causes large amount of computation.To avoid this case,many scholars present some inexact Newton methods,which computes linear subproblems approximatively.Recently,Fan and Yuan[4]proposed a regularization exact Newton method,which need exactly compute two linear subproblems each iteration.In this paper,we extend the exact method in[4]to inexact case.The paper is organized as follows:In Chapter 1,we simply introduce the background and related preparatory knowledge.In Chapter 2,we introduce a regularization inexact Newton method for solving monotone nonlinear equations and discusss its convergence property Under suitable conditions,we show that the proposed method has locally quadratic convergence rate.In Chapter 3,we present a global version of the proposed method in Chapter 2 by the use of some line search.We prove that this new version has global convergence property.Moreover,we do some numerical experiments.Numerical results show that the method is efficient.