A Generalized Newton Method And Its Convergence For Absolute Value Equation

Absolute value equation(AVE) Ax-|x|=b,A∈R^n×n,b∈Rⁿ is a special class of nonlinear equations and it is NP-Hard.The study of the AVE is inspired from the interval linear equations and the well-known linear complementarity problem(LCP), which is a uniform framework for many mathematical programs.The AVE is shown to be equivalent to the LCP.In this thesis we mainly study the solving of AVE.Based on its semismoothness,a generalized Newton method which combines the semismooth and the smoothing Newton steps is proposed.We establish global and finite convergence of the method under mild conditions.Preliminary numerical results indicate that the generalized Newton method is promising.This thesis is organized as follows.In Chapter 1,the background and the present improvements of AVE are introduced, some effective methods are briefly analyzed here.In Chapter 2,some basic conceptions and theorems are given,which are useful in the following.We present the equivalence between AVE and LCP,as well as the relationship between AVE and linear interval equations.It is worth mentioning that the interval matrix[A-I,A+I]is regular is weaker than that the singular values of A exceed 1.At last,we establish the generalized Newton method.In Chapter 3,the convergence of the generalized Newton method is studied and numerical experiments are given.Under the condition that the interval matrix[A-I, A+I]is regular,the generalized Newton method converge to the unique solution of AVE globally and finitely.Numerical results show that the generalized Newton method perform better than semismooth Newton method and smoothing Newton method.In Chapter 4,the main conclusions of this thesis are summarized.