^{1}

^{*}

^{1}

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.

The linear complementarity problem, denoted by

where

There exist several methods for solving

In [

Find

where

The nonlinear penalized problems (1.2) corresponding to the linear complementarity problem (1.1), which its research has achieved good results. In 1984, Glowinski [

Although the studies solving for the linear complementarity problem based on the nonlinear penalized equation have good results. But there is no method that is given for solving the nonlinear penalized equation. Throughout the paper, we propose a generalized Newton method for solving the nonlinear penalized equation with under the suppose

Some words about our notation:

The definition of interval matrix arises from the linear interval equations [

Lemma 1: Assume that interval matrix

Proof: By definition of

By the assumptions, we have

Lemma 2: Let

Definition 1 [

Lemma 3 [

In this section, we will propose that a new generalized Newton method based on the nonlinear penalized equation for solving the linear complementarity problem. Because when

These case penalty problems for the continuous Variational Inequality and the linear complementarity problems are discussed in [

Let us note

Thus, nonlinear penalized equation (3.1) is equivalent to the equation

A generalized Jacobian

where

depending on whether the corresponding component of

Then

Since

Proposition 1

Proof: Since

By [

Let

Proposition 2

Proof: Since

lation between the (3.1) and the LCP (3.5), we can easily deduce that the (3.1) is uniquely solvable for any

Algorithm 3

Step 1: Choose an arbitrary initial point

Step 2: for the

Step 3: If

Step4: If

We will show that the sequence

Theorem 3: Suppose that the interval matrix

Proof: Suppose that sequence

where

We know subsequence

Letting

Since

Under a somewhat restrictive assumption we can establish finite termination of the generalized Newton iteration at a penalized equation solution as follows.

Theorem 4: Suppose that the interval matrix

Proof: Suppose that

since

Letting

So the iteration (3.6) linearly converges to a solution

In here, we will focus on the convergence of Algorithm 3.

Theorem 5: Suppose M is P-Matrix and that the interval matrix

Proof: Since M is P-Matrix, then the

let

we have

In this section, we give some numerical results in order to show the practical performance of Algorithm 2.1. Numerical results were obtained by using Matlab R2007 (b) on a 1G RAM, 1.86 Ghz Intel Core 2 processor. Throughout the computational experiments, the parameters were set as

Example 1: The matrix

The computational results are shown in

Example 2: The matrix

The computational results are shown in

6 | 3 | 2 | ||

6 | 3 | 2 | ||

7 | 3 | 2 | ||

7 | 3 | 2 |

6 | 3 | 2 | ||

6 | 3 | 2 | ||

8 | 3 | 2 | ||

8 | 3 | 2 | ||

16 | 3 | 2 | ||

16 | 3 | 2 |

This work supported by the Science Foundation of Inner Mongolia in China (2011MS0114)

Hai-ShanHan,Lan-Ying , (2015) An Interval Matrix Based Generalized Newton Method for Linear Complementarity Problems. Open Journal of Applied Sciences,05,443-449. doi: 10.4236/ojapps.2015.58044