Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints

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.


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: where : is twice continuously differential real-valued function; are given matrices; b and q 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 Section 3, the algorithm is proposed.

Preliminaries
For reader's convenience, we use following notation throughout this paper: , , , , , , , , , 1, 2, , , : 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: , 0, 0, 0, 0,

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 ( ) G Ω at Ω .For an approximate solution k z of (2.6), in order to produce an improving direction, we consider the following system of linear equations ( ) where ( ) According to the constraint conditions, we now find a step length for current iterated point.First, we consider computing the step length of ( ) , , , x y w λ .In the first place, for each constraint in (2.7), we should use the k Ω and k d to computer a step length: 1, 0, min 1, max 0, , 0.
where k x d is the element of k d .Similar to the discussion of step length about x, we can obtain the step length  Secondly, we will consider the step length of ( ) Now we present the algorithm.

Algorithm A:
Step 0: Given a feasible initial point Ω , let 1 k = ; Proof.From the construction of the algorithm, we have k z F ∈ for sufficient large k and z F ∈ .And because the MPEC-LICQ holds on z , then z is a B-stationary point of problem (1.1).
, because it will be convenient for our computing.

Lemma 3 . 1 .
The coefficient matrix of (L − M) is positive definite, and furthermore, (L − M) method has unique solution.
α as its variable, then j α is as follows:

2 G 1 .
Ω < , then stop; else get the k d for (3.1);Step 2: Compute the step length k θ ; Suppose that Ω is generated by Algorithm A and converges to Ω ; if k z F ∈ for infinitely many k, let the MPEC-LICQ hold on z , then z is a B-stationary point of problem (1.1).
the feasible set of problem (1.1).Now we give two definitions as follow.
Definition 2.1.Let * z be a feasible point of MPLCC (1.1), we say that MPEC linear independence constraint qualification is satisfied at * z if the gradient vectors ( ) ( ) 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 µ is set to 1.