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