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In this paper, a new method for solving a mathematical programming problem with linearly complementarity constraints (MPLCC) is introduced, which applies the Levenberg-Marquardt (L-M) method to solve the B-stationary condition of original problem. Under the MPEC-LICQ, the proposed method is proved convergent to B-stationary point of MPLCC.

The mathematical program with equibrium constraints (MPEC) has extensive application in area engineering design and economic model [

where

Complementarity constraints in MPEC are known to be difficult to treat. Research work on the MPEC includes the monograph of Luo et al. [

The plan of the paper is as follows: in Section 2, some preliminaries and model we used are presented; in Sec- tion 3, the algorithm is proposed.

For reader’s convenience, we use following notation throughout this paper:

Let F denote the feasible set of problem (1.1).

Now we give two definitions as follow.

Definition 2.1. Let

is linearly independent, where

Definition 2.2. Under the MPEC-LICQ, a feasible point z is a B-stationary of problem (1.1) if there exist multiplier vectors

As we know, most of the works on MPLCC want to get the B-stationary point of problem (1.1), so we also put emphasis on trying to construct a method to obtain the B-stationary of MPLCC (1.1). Now we rewrite the conditions (2.1)-(2.5) in term of lagrange multipliers as follow:

subject to:

and

where

Remark: In (2.7) we replace

Without any reformulation and relaxing techniques, we now use L-M method to solve the nonlinear systems (2.6). Firstly, let J be the Jacobian of

where

Lemma 3.1. The coefficient matrix of (L − M) is positive definite, and furthermore, (L − M) method has unique solution.

According to the constraint conditions, we now find a step length for current iterated point. First, we consider computing the step length of

where

As to calculating the step length for the constraint

so

Secondly, we will consider the step length of

In this paper, we take

Lemma 3.2. Let

Proof. In view of Equation (3.1) and the positive definition of matrix

Now we present the algorithm.

Algorithm A:

Step 0: Given a feasible initial point

Step 1: If

Step 2: Compute the step length

Step 3:

Theorem 3.1. Suppose that

many k, let the MPEC-LICQ hold on

Proof. From the construction of the algorithm, we have

This work was supported in part by the National Natural Science Foundation (No. 11361018), the Natural Science Foundation of Guangxi Province (No. 2014GXNSFFA118001), Key Program for Science and Technology in Henan Education Institution (No. 15B110008) and Huarui College Science Foundation (No. 2014qn35) of China.

CongZhang,LiminSun,ZhibinZhu,MingleiFang, (2015) Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints. American Journal of Computational Mathematics,05,239-242. doi: 10.4236/ajcm.2015.53020