Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints ()

Cong Zhang^{1*}, Limin Sun^{1}, Zhibin Zhu^{2}, Minglei Fang^{3}

^{1}Department of Mathematics and Computer Science, Huarui College, Xinyang Normal University,
Xinyang, China.

^{2}School of Mathematics & Computational Science, Guilin University of Electronic Technology, Guilin, China.

^{3}College of Science, Anhui University of Science and Technology, Huainan, China.

**DOI: **10.4236/ajcm.2015.53020
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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.

Keywords

Mathematical Programs with Linear Complementarity Constraints, MPEC-LICQ, B-Stationarity, Levenberg-Marquardt Method

Share and Cite:

Zhang, C. , Sun, L. , Zhu, Z. and Fang, M. (2015) Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints. *American Journal of Computational Mathematics*, **5**, 239-242. doi: 10.4236/ajcm.2015.53020.

1. Introduction

The mathematical program with equibrium constraints (MPEC) has extensive application in area engineering design and economic model [1] . It has been an active research topic in recent years. In this paper, we consider the mathematical programming problem with linearly complementarity constraints (MPLCC), which is a special case of the MPEC:

(1.1)

where is twice continuously differential real-valued function;, and are given matrices; and are given p, m dimensional vectors, respectively.

Complementarity constraints in MPEC are known to be difficult to treat. Research work on the MPEC includes the monograph of Luo et al. [1] in which Bouligand stationary condition is introduced that provides a comprehensive study on MPEC. Based on different formulations, there are many algorithms such as Fukushima [2] , Zhu [3] , Zhang [4] [5] , Jiang [6] , Tao [7] , and Jian [8] . Notice that B-stationary condition is a stronger stationary point. Differing from the approaches mentioned above, we directly introduce L-M technique, without any reformulation or relax form, to solve the B-stationary condition of MPLCC (1.1).

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.

2. Preliminaries

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 be a feasible point of MPLCC (1.1), we say that MPEC linear independence constraint qualification is satisfied at if the gradient vectors

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, and such that

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

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:

(2.6)

subject to:

(2.7)

and

(2.8)

where,.

Remark: In (2.7) we replace with, because it will be convenient for our computing.

3. The Description of Algorithm

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 at. For an approximate solution of (2.6), in order to produce an improving direction, we consider the following system of linear equations

(3.1)

where, is a constant.

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. In the first place, for each constraint in (2.7), we should use the and to computer a step length:

(3.2)

where is the element of. Similar to the discussion of step length about x, we can obtain the step length about.

As to calculating the step length for the constraint we get the solution to the equation with as its variable, then is as follows:

(3.3)

so

Secondly, we will consider the step length of. Based on the step length that we obtain above, we can compute the value of. If there is some i that, then the step length of corres- ponding variables is obtained by the same way in (3.2) in order to satisfy the constraints (2.8); otherwise the step lengths of u, v are set to 1. The step length of is set to 1.

In this paper, we take as the merit function.

Lemma 3.2. Let be computed from (3.1), then

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

Now we present the algorithm.

Algorithm A:

Step 0: Given a feasible initial point, let;

Step 1: If, then stop; else get the for (3.1);

Step 2: Compute the step length;

Step 3:, go to Step 1, where.

Theorem 3.1. Suppose that is generated by Algorithm A and converges to; if for infinitely

many k, let the MPEC-LICQ hold on, then is a B-stationary point of problem (1.1).

Proof. From the construction of the algorithm, we have for sufficient large k and. And because the MPEC-LICQ holds on, then is a B-stationary point of problem (1.1).

Funding

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.

NOTES

^{*}Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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[2] |
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