Algorithms And Convergence Analysis For Absolute Value Equations

The study of the absolute value equations(AVE)Ax|x|=b, A∈n×n, bn∈is inspired from the linear complementary problem(LCP). The AVE is a spe-cial class of nonlinear equations. The absolute value equations is equivalent tolinear complementary problem and bilinear programming problem, therefore, lin-ear programming, quadratic programming, linear complementary and so on canbe transformed equivalently to absolute value equations. So the absolute valueequations have strong application background.We mainly give a research on solving absolute value equations in this paper.According to the nonsmoothing of the absolute value, we propose a smoothingNewton method and a negative gradient descent algorithm for solving the absolutevalue equations. And we prove the feasibility and the convergence of algorithms.Preliminary numerical results show that the two algorithms are promising. Thefull thesis is divided into four chapters.The first chapter is the introduction. We describe the application back-grounds, the research situations and achievements of the absolute value equationsproblem. Several efective algorithms and the research ideas are analyzed in thischapter.In the second chapter,we directly give a smoothing function of the absolutevalue equations. And then,under the condition that the interval matrix [A I, A+I] is regular, we propose a smoothing Newton method for solving AVE.The convergence of the algorithm is proved.In the third chapter, we transform the AVE into a semismooth functionequations Φ(x)=0based on Fischer-Burmeister function and generalized lin-ear complementary problem. And then, we give a smoothing function Φ (x) ofΦ(x), and also discuss the basic properties of it. Then we propose an imporved smoothing Newton method for solving AVE. Under the condition that the inter-val matrix [A—I, A+I] is regular, the imporved algorithm is global superlinear convergence to the unique solution of the absolute value equations. Numerical results of this chapter show that the algorithm is feasible.In Chapter4, in order to avoid the non-differentiability of the absolute value function, we transform the absolute value equations into a unconstrained opti-mization problem. Then we propose a negative gradient descent algorithm. The feasibility and convergence of the algorithm are also proved. Numerical results show that the algorithm is feasible.