An Interval Matrix Based Generalized Newton Method for Linear Complementarity Problems

Hai-Shan Han,Lan-Ying

doi:10.4236/ojapps.2015.58044

Abstract

The penalty equation of LCP is transformed into the absolute value equation, and then the existence of solutions for the penalty equation is proved by the regularity of the interval matrix. We propose a generalized Newton method for solving the linear complementarity problem with the regular interval matrix based on the nonlinear penalized equation. Further, we prove that this method is convergent. Numerical experiments are presented to show that the generalized Newton method is effective.

Highlights

The linear complementarity problem, denoted by LCP (M, q), is to find a vector x ∈ Rn such that x ≥ 0, Mx − q ≥ 0, xT (Mx − q) =0 (1.1)where M ∈ Rn×n is a given matrix and q ∈ Rn is a given vector
It is shown that the solution to the penalized equation converges to that of the linear complementarity problem with matrix is positive definite
Throughout the paper, we propose a generalized Newton method for solving the nonlinear penalized equation with under the suppose [M, M + λI ] is regular

Summary

Introduction

The linear complementarity problem, denoted by LCP (M , q) , is to find a vector x ∈ Rn such that x ≥ 0, Mx − q ≥ 0, xT (Mx − q) =0. It is shown that the solution to the penalized equation converges to that of the linear complementarity problem with matrix is positive definite. In 2008, Yang [7] proved that solution to this penalized Equations (1.2) converged to that of the LCP at an exponential rate for a positive definite matrix case where the diagonal entries were positive and off-diagonal entries were not greater than zero. The studies solving for the linear complementarity problem based on the nonlinear penalized equation have good results. Throughout the paper, we propose a generalized Newton method for solving the nonlinear penalized equation with under the suppose [M , M + λI ] is regular. Numerical experiments are given to show the effectiveness of the proposed method

Preliminaries

Generalized Newton Method

The Convergence of the Algorithm

Numerical Experiments