A Penalized-Equation-Based Generalized Newton Method for Solving Absolute-Value Linear Complementarity Problems

We consider a class of absolute-value linear complementarity problems. We propose a new approximation reformulation of absolute value linear complementarity problems by using a nonlinear penalized equation. Based on this approximation reformulation, a penalized-equation-based generalized Newton method is proposed for solving the absolute value linear complementary problem. We show that the proposed method is globally and superlinearly convergent when the matrix of complementarity problems is positive definite and its singular values exceed 1. Numerical results show that our proposed method is very effective and efficient. 1. Introduction Let be a given function. The complementarity problems, CP( ) for short, is to find a solution of the system The CP( ) is called the linear complementarity problems (for short LCP) if is an affine mapping of the form where and . Otherwise, the CP( ) is called the nonlinear complementarity problems (NCP( )). The systematic study of the finite-dimensional CP( ) began in the mid-1960s; in a span of five decades, the subject has developed into a very fruitful discipline in the field of mathematical programming. The developments include a rich mathematical theory, a host of effective solution algorithms, a multitude of interesting connections to numerous disciplines, and a wide range of important applications in engineering and economics (see, e.g., [1–4] and the references therein). Generalized Newton method (semismooth Newton method) is one of efficacious algorithms for solving CP( ). The main idea of semismooth Newton method is based on an equivalent reformulation of the complementarity problems consisting of a nonsmooth equation and then solving the nonsmooth equation by Newton type method (see, e.g., [5, 6]). Most reformulations of the CP( ) are based on the Fischer-Burmeister function [7] (see, e.g., [8–10] and the references therein). Chen et al. [11] introduced a penalized Fischer-Burmeister function and proposed a new semismooth Newton method based on this new NCP function. Kanzow and Kleinmichel [12] proposed a new, one-parametric class of NCP functions based on Fischer-Burmeister function and gave a semismooth Newton method via these NCP functions. Kanzow [13] researched an inexact semismooth Newton method based on Fischer-Burmeister function and penalized Fischer-Burmeister function. Ito and Kunisch [14] studied a semismooth Newton method based on the max-type NCP function. All generalized Newton methods mentioned above involve the continuously differentiable assumption on in CP( ). The existed generalized Newton