R. Zhu et al. (Eds.): ICICA 2010, Part I, CCIS 105, pp. 390–398, 2010. © Springer-Verlag Berlin Heidelberg 2010

Iterative Method for a Class of Linear Complementarity Problems

Longquan Yong

Department of Mathematics, Shaanxi University of Technology, Hanzhong 723001, Shaanxi, P.R. China

yonglongquan@sohu.com

Abstract. An iterative method for solving a class of linear complementarity problems with positive definite symmetric matrices is presented. Firstly, linear complementarity problem is transformed into absolute value equation, which is also a fixed-point problem. Then we present an iterative method for the linear complementarity problem based on fixed-point principle. The method begins with an initial point chosen arbitrarily and converges to optimal solution of original problem after finite iterations. The effectiveness of the method is demonstrated by its ability to solve some standard test problems found in the literature.

Keywords: iterative method, linear complementarity problem, positive definite symmetric matrices, absolute value equation, fixed-point principle.

1 Introduction

The linear complementarity problem (LCP) is to determine a vector pair ( , )zω

satisfying,

,0 , 0 ,

0 ,T

M z qz

z

ωωω

⎧⎪⎨⎪⎩

− =≥ ≥

= (1)

where ,, , n nn M Rz q Rω ×∈∈ and M is a positive definite symmetric

matrix. LCP (1) is a fundamental problem in mathematical programming. It is known that any differentiable linear and quadratic programming can be formulated into LCP (1). LCP also has wide range of applications in economic and engineering. The interested readers are referred to the survey paper [1].

A number of direct methods have been proposed for their solution. The book by Cottle et al. [2] is a good reference for pivoting methods developed to solve LCP. Another important class of methods used to tackle LCP is the interior point methods. Interior point methods (IPMs) are an important method for LCP. Modern interior point methods originated from an algorithm introduced by Karmarkar in 1984 for

Iterative Method for a Class of Linear Complementarity Problems 391

linear programming [3]. Most IPMs for LCP can be viewed as natural extensions of the interior point methods for linear programming. The most successful interior point method is the primal-dual method. The primal-dual IPMs for linear optimization problem were first introduced in [4] and [5]. Kojima et al. [4] first proved the polynomial computational complexity of the algorithm for linear optimization problem, and since then many other algorithms have been developed based on the primal-dual strategy. Kojima et al. [6] proposed a polynomial time algorithm for monotone LCP under the nonemptiness assumption of the set of feasible interior point.

Each algorithm in the class of interior point methods for the LCP has a common

feature such that generates a sequence {( , ), 0,1,2, }k kz kω = L in the positive orthant

of 2nR under the assumption of knowing a feasible initial point 0 0( , )zω . If each

point ( , )k kzω of the generated sequence satisfies the equality system Mz qω − = ,

then we say that the algorithm is a feasible interior point algorithm. However, it is a very difficult task in finding a feasible initial point to start the

algorithm. To overcome this difficulty, recent studies have been focused on some new interior point algorithms without the need to find a feasible initial point. In 1993, Kojima et al. presented the first infeasible interior point algorithm with global convergence [7], soon later Zhang [8] and Wright [9] introduced this technique to the linear complementarity problem. Thus few numerical experiments were reported about the feasible interior point algorithm because of difficulty in selecting the strictly feasible starting point.

Based on above reasons, in this paper, we proposed an iterative method for solving LCP (1) with positive definite matrices. Firstly, LCP (1) is transformed into absolute value equation, which is also a fixed-point problem. Then we present an iterative method for the LCP (1) based on fixed-point principle. The method begins with an initial point chosen arbitrarily and converges to optimal solution of original problem after finite iterations. The effectiveness of the method is demonstrated by its ability to solve some standard test problems found in the literature.

In section 2 we transform LCP (1) problem into fixed-point problem and present our new algorithm. The proof of the convergence result will be developed in section 3. Section 4 gives some standard test problems found in the literature. Section 5 contains some concluding remarks and comments.

We now describe our notation. All vectors will be column vectors. For nx R∈ the 2-norm will be denoted by x , while x will denote the vector with absolute values

of each component of x . The notation m nA R ×∈ will signify a real m n× matrix. If

A is square matrix of order n , its norm, dented by A is defined to be the

supremum of { }|| || / || ||: , 0nAx x x R x∈ ≠ . From this definition, we have

Ax A x≤ for all nx R∈ . We write I for the identity matrix ( I is suitable

dimension in context). A vector of zeros in a real space of arbitrary dimension will be denoted by 0.

392 L. Yong

2 Theoretical Background and New Algorithm

The following result is due to W. M. G. Van Bokhoven [10]. We consider the LCP (1) where M is assumed to be a positive definite symmetric matrix, for 0q ≥ ,

0zω = = is the unique solution of the LCP (1). So we only consider the case

q Q∉ , here { 0}Q q q= ≥ .

Theorem 1 (W. M. G. Van Bokhoven). Let M be positive definite and symmetric. The LCP (1) is equivalent to the fixed-point problem of determining nx R∈ satisfying

( )x f x= . (2)

where ( ) ( ) ( )1 1, ,( ) c I M I M c I M qf x B x B

− −+ + − = − += = .

Proof. In (1) transform the variables by substituting

, ,j j j j j jx x x xzω = − = + for each 1 to j n= (3)

We verify that the constraints 0 0,j jzω ≥ ≥ for 1 to j n= automatically hold, from

(3). Also substituting (3) in M z qω − = , lead to ( ) 0f x x− = . Further, 0j jzω =

for 1 to j n= , by (3). So any solution x of (2) automatically leads to a solution of

the LCP (1) through (3). Conversely suppose ( , )zω is the solution of LCP (1).

Then ( ) / 2x z ω= − can be verified to be the solution of (2).

Since M is positive definite and symmetric, all its eigenvalues are real and positive. If 1, , nλ λL are the eigenvalues of M , then the eigenvalues of

( ) ( )1I M I MB

−+ −= are given by 1(1 ) (1 )i i iμ λ λ−= + − , 1 to i n= ; and hence

all iμ are real and satisfy 1iμ < for all i (since 0iλ > ). Since B is also symmetric

we have { : 1 to } 1iB max i nμ= = < .

Following we describe our new method for solving absolute value equations (2). Since (2) is also a fixed-point problem, iterative method is a common method for solving fixed-point problem.

The name iterative method usually refers to a method that provides a simple formula for computing the ( 1)thk + point as an explicit function of the thk point

1 ( )k kx f x+ = . The method begins with an initial point 1x (often 1x can be chosen

arbitrarily, subject to some simple constraints that may be specified, such as 1 0x ≥ ,

etc.) and generates the sequence of points 1 2{ , , }x x L one after the other using the

above formula. The method can be terminated whenever one of the points in the sequence can be recognized as being a solution to the problem under consideration. If finite termination does not occur, mathematically the method has to be continued indefinitely. In some of these methods, it is possible to prove that the sequence { }kx

Iterative Method for a Class of Linear Complementarity Problems 393

converges in the limit to a solution of the problem under consideration, or it may be possible to prove that every accumulation point of the sequence { }kx is a solution of

the problem. In practice, it is impossible to continue the method indefinitely. In such cases, the sequence is computed to some finite length, and the final solution accepted as an approximate solution of the problem.

Most of the algorithms for solving nonlinear programming problems are iterative in nature and the iterative method discussed here can be interpreted as specializations of some nonlinear programming algorithms applied to solve AVE (1).

Based on above discussion, we now state iterative method for solving (2).

Algorithm 1 (The iterative method) Given arbitrary initial point 1 nx R∈ , convergence tolerance 0ε > ; For 1, 2,3,k = L

Calculate the next point point

1 ( )k k k cf x B xx + = += ; (4)

End.

The equation (4) defines the iterative scheme. Beginning with the initial point 1 nx R∈

chosen arbitrarily, generate the sequence 1 2{ , , }x x L using (4) repeatedly. This

iteration is just the successive substitution method for computing the Brouwer's fixed-point of (2). We will now prove that the sequence generated 1 2{ , , }x x L converges in

the limit to the unique fixed point *x of (2).

3 Convergence Analysis

Theorem 2. When M is positive definite and symmetric. The sequence of points { }kx defined by (4) converges in the limit to *x , the unique solution of (2), and the

solution * *( , )zω of the LCP (1) can be obtained from the transformation (3).

Proof. For any , nx y R∈ ，we have

( ) ( )( )( )x f y B x y B x y x yf − = − ≤ − ≤ − ,

since ( )x y x y− ≤ − and 1B < . So ( )f x is a contraction mapping (see

reference [11]) and by Banach contraction mapping theorem the sequence { }kx

generated by (4) converges in the limit to the unique solution *x of (2). The rest follows from Theorem 1.

We will denote B by the symbol ρ . We known that 1ρ < , and it can actually

be computed by matrix theoretic algorithms (see reference [12]).

Theorem 3. When M is positive definite and symmetric. Let kx be the thk point

obtained in the iterative scheme (4) and let *x be the unique solution of (2). Then for 1k ≥ ,

394 L. Yong

1 * 2 1

1

kkx x x x

ρρ

+ − ≤ −−

.

Proof. We have 1 * * *( ) ( )k k kx x f x f x x xρ+ − = − ≤ − .

Applying the iterative scheme (4) repeatedly we get

1 * 1 *k kx x x xρ+ − ≤ − . (5)

And for 2k ≥ we have 1 1 2 1k k kx x x xρ+ −− ≤ − using iterative scheme (4)

repeatedly.

We also have * 1 * 2 2 1( )x x x x x x− = − + − . So we have * 1 * 2 2 1x x x x x x− ≤ − + − . Using

this same argument repeatedly, and the fact that the * lim kx x= as k tend to ∞ , we get

* 1 1 2 1 2 1

1 0

1

1k k k

k kx x x x x x x xρ

ρ∞ ∞+

= =− ≤ − ≤ − ≤ −

−∑ ∑ . (6)

Using formula (6) in (5) leads to 1 * 2 1

1

kkx x x x

ρρ

+ − ≤ −−

for 1k ≥ .

Corollary 1. When M is positive definite and symmetric. Let 1 0x = . We have

1 *

1

kkx x c

ρρ

+ − ≤−

.

Proof. Follows from Theorem 3.

Theorem 4. When M is positive definite and symmetric. Let 1 0x = . Then the

algorithm terminates with an approximate solution in ( )1log( ) logO cρε ρ−

iterations.

Proof. Since 1 1 2 1 1 2 1 1 1( )k k k k k kx x x x x f x cρ ρ ρ ρ+ − − − −− ≤ − = = = ,

A sufficient condition for 1k kx x ε+ − ≤ is given by 1k cρ ε− ≤ .

This implies that ( 1) log log logk cρ ε− + ≤ .

So we obtain the bound 1

log( ) logck ρε ρ−≥ .

Thus the above theorem is proven.

Corollary 2. When M is positive definite and symmetric. Let *x is the unknown

solution of (2). Then * 1

1x c

ρ≥

+.

Iterative Method for a Class of Linear Complementarity Problems 395

Proof. From (2), we have

( ) ( )* * * *x B x c c B x c xρ= + ≥ − ≥ − .

So

* 1

1x c

ρ≥

+.

4 Numerical Results

LCP1. First we consider one LCP problem where the data (M, q) are

3 2 1 14

2 2 1 , 11 .

1 1 1 7

M q

− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

Since M is symmetric and its eigenvalues eig(M)=[0.3080,0.6431,5.0489], the LCP is uniquely solvable by theorem 2. Chose initial point 1 0x = , 41 10ε −= × . We use

1k kx x ε+ − ≤ as the stopping rule. After 13 iterations, the unique solution to

fixed-point problem (2) is * [-1.5000,2.0000,1.5000]Tx = . Thus the unique solution * *( , )zω of the LCP (1) can be obtained from the transformation (3), that is

* * * 2 .9999,0,0[ ]Tx xω = − = , * * * 0,3.9999,3.0000[ ]Tx xz = + = .

LCP2. This test problem was taken from [13] and has also been cited in [14]. It is a standard test problem for LCP,

1 2 2 2 1

2 5 6 6 1

, .2 6 9 10 1

2 6 10 4 3 1

M q

n

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

L

L

L

M M M O M M

L

Where the matrix M is positive definite, the solution is * (1,0, ,0)Tx = L . Using the

Algorithm 1, the number of iterations to converge to the optimal solutions is given in the Table 1.

LCP3. This problem was also taken from [15], M is the triple diagonal matrix

4 1 0 0 1

1 4 1 0 1

, .0 1 4 0 1

0 0 0 4 1

M q

− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

L

L

L

M M M O M M

L

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Table 1. The number of iterations of LCP2 by Algorithm 1

Dimension Iterations Elapsed time (in seconds)

4 49 0

8 227 0

16 832 0.0470

32 2705 0.8440

64 7747 27.2500

128 17731 297.4690

Table 2. The number of iterations of LCP3 by Algorithm 1

Dimension Iterations Elapsed time (in seconds)

4 17 0

8 17 0

16 17 0

32 17 0

64 17 0

128 17 0

Table 3. The number of iterations of LCP4 by Algorithm 1

Dimension Iterations Elapsed time (in seconds)

4 38 0

8 85 0

16 255 0

32 573 0.0630

64 1385 0.5310

128 2501 5.5310

It is a standard test problem for LCP, too. Using the Algorithm 1, the number of

iterations to converge to the optimal solutions is given in the Table 2.

LCP4. Following we consider some randomly generated LCP with positive definite and symmetric M where the data (M, q) are generated by the Matlab scripts:

n=input('dimension of matrix M'); rand('state',0); R=rand(n,n); M=R'*R+n*eye(n); q=rand(n,1);

Iterative Method for a Class of Linear Complementarity Problems 397

and we set the random-number generator to the state of 0 so that the same data can be regenerated. Choose initial point 1 0x = , 41 10ε −= × . In all instances the algorithm perform extremely well, and finally converge to an optimal solution for the LCP. More detail of numerical results are presented in Table 3.

All the experiments were performed on Windows XP system running on a Hp540 laptop with Intel(R) Core(TM) 2×1.8GHz and 2GB RAM, and the codes were written in Matlab 6.5.

5 Conclusion

In this work we have established an iterative method for solving a class of linear complementarity problems with positive definite symmetric matrices. We have established global convergence for this method. Preliminary numerical experiments with standard test problems and some randomly generated problems indicate that the proposed algorithm seems promising for solving the LCP. Acknowledgments. The author is very grateful to the referees for their valuable comments and suggestions. This work is supported by Natural Science Foundation of Shaanxi Educational Committee (No.09JK381).

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