Algorithms And Convergence For Absolute Value Equations

Absolute value equations(AVE) Ax x b， A R n n,b Rnis a special class ofnonlinear equations. The study of the AVE is inspired from the interval linear equationsand the well-known linear complementarity problem(LCP). Because of its equivalenceto the LCP, many math problems such as linear programming and quadraticprogramming can be transformed into the AVE. So the absolute value equations havestrong application background.We mainly discuss the algorithms on solving absolute value equations in this paper.In the condition that the singular values ofAexceed1, a smoothing Newton algorithmwith value function is proposed based on its semis-moothness firstly. Then because theabsolute value equations is proved to equal to an unconstrained minimization problem, aglobal interation algorithm is introduced naturally. Theoretic provings and numericalresults both show the validity and effectiveness of this two algorithms.This paper points five chapters, its structure is as follows.The first chapter is the introduction. We describe backgrounds and research situ-ations about the absolute value equations. Theories and several existing algorithms aregiven in this chapter.In the second chapter，it is proved that theAVE is equal to the BLP and LCP. Thenthe famous Alternative theorems is given.In the third chapter, we construct a smoothing Newton algorithm by directly givinga smoothing function in the condition that the singular values of A exceed1. Then theconvergence is proved and numerical results are given.In the forth chapter, by using the maximum entropy function, the other smoothingadvanced function of absolute value equations is got, and then a Newton method andconvergence are given. Numerical results also indicate that this method is feasible andeffective.In the last chapter, the absolute value equations can be proved to equal to anunconstrained minimization problem in Noor’s theory. Then specific iterative steps andits convergence are given. Numerical results also show this method is feasible.