^{1}

^{1}

The existence condition of the solution of special nonlinear penalized equation of the linear complementarity problems is obtained by the relationship between penalized equations and an absolute value equation. Newton method is used to solve penalized equation, and then the solution of the linear complementarity problems is obtained. We show that the proposed method is globally and superlinearly convergent when the matrix of complementarity problems of its singular values exceeds 0; numerical results show that our proposed method is very effective and efficient.

Given a matrix

is called the linear complementarity problem (LCP). We call the problem the LCP (A, b). It is well known that several problems in optimization and engineering can be expressed as LCPs. Cottle, Pang, and Stone [

There are a large number of general purpose methods for solving linear complementarity problems. We can divide these methods into essentially two categories: direct methods, such as pivoting techniques [

The penalty method has been used an LCP (or, equivalently, a variational inequality) [

Find

where

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

approach to the linear complementarity problem. For the penalty Equation (1.2) Li [

Some words about our notation: I refers to the identity matrix, and ^{T} refers to the transpose of the y, we denote by

In this section, we will propose that a new generalized Newton method based on the nonlinear penalized Equation (1.2) for solving the linear complementarity problem.

Proposition 1 [

Proposition 2.

Proof: Since the singular values of A exceed 0, then A is a positive definite matrix，and

Let us note

Thus, nonlinear penalized Equation (1.2) is equivalent to the equation

A generalized Jacobian

where

equavelently

Algorithm 1

Step 1: Choose an arbitrary initial point

Step 2: for the

Step 3: If

Step4: If

We will show that the sequence

cumulation point

Theorem 1: Suppose the singular values of M exceed 0. Then, the sequence

Proof. Suppose that sequence

where

We know subsequence

Letting

Since the singular values of A exceed 0, then A is regular, and

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

Theorem 2: Suppose the singular values of A exceed 0 and

ciently large

Proof. Similar to the proof of Theorem 4 in [

Theorem 3: Suppose the singular values of A exceed 0 and

1 linearly converges from any starting point

Proof. Similar to the proof of Theorem 5 in [

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 A of linear complementarity problem

n | x^{0} | k | m | |
---|---|---|---|---|

6 | 3 | 2 | ||

6 | 3 | 2 | ||

7 | 3 | 2 | ||

7 | 3 | 2 |

x^{0} | k | m | |
---|---|---|---|

1 | 26 | ||

2 | 26 | Results are as above. | |

2 | 26 | Results are as above. |

n | x^{0} | k | m | |
---|---|---|---|---|

6 | 3 | 2 | ||

6 | 3 | 2 | ||

8 | 3 | 2 | ||

8 | 3 | 2 | ||

16 | 3 | 2 | ||

16 | 3 | 2 |

The computational results are shown in

Example 2: The matrix A of linear complementarity problem

Optimal solution of this problem is

Example 3: The matrix A of linear complementarity problem

The computational results are shown in

HaishanHan,YuanLi, (2015) A Singular Values Based Newton Method for Linear Complementarity Problems. Applied Mathematics,06,2354-2359. doi: 10.4236/am.2015.614207