2017 2nd International Conference on Computational Modeling, Simulation and Applied Mathematics (CMSAM 2017) ISBN: 978-1-60595-499-8
Multisplitting Iterative Methods with General Weighting Matrices for
Solving Symmetric Positive Linear Complementarity Problem
Ban-xiang DUAN and Ai-min YU
Computer Engineering Technical College, Guangdong Institute of Science and Technology, Zhuhai, Guangdong 519090, China
Keywords: Linear complementarity problem, Symmetric positive definite matrix, Weighting matrices,
Multisplitting, Convergence.
Abstract. In this paper, by making use of optimal models, we study the weighting matrices of the multisplitting parallel methods for solving the symmetric positive definite linear complementarity problem, which is a powerful alternative for solving the large sparse linear complementarity problems. In our multisplitting there is only one that is required to be P-regular splitting and all the others can be constructed arbitrarily, which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix, but also relaxes the constraints to the weighting matrices (unlike the standard methods, they are not necessarily nonnegative diagonal scalar matrices or given in advance). Finally, we prove the convergence of this new method.
Introduction
This paper focuses on thelinear complementarity problem, which is to find a pair of real vectors r
and zRn such that
0 ) (
, 0 , 0
:qAz z z qAz
r T (1)
Where ARnn and qRn are given real matrix and vector, respectively, and zT denotes the transpose of the vector z.
This problem has been intensely studied since the 1960s. Many alternative formulations have been investigated, and efficient algorithms have been proposed in Cottle et al. (1992)[1]. Much attention has recently been paid on a class of iterative methods called the matrix-splitting method [2-4]. Matrix splitting method for linear complementarity problem exploits particular features of matrices such as the sparsity and the block structure. Such an approach is motivated by matrix splitting methods for the linear complementarity problem.
Recently, Wen present a multisplitting parallel methods for solving the symmetric positive definite linear systems[5].This method is based on the matrix multisplitting technique introduced by O’Leary and White[6] for the system of linear equations.
In this paper, the multisplitting parallel methods for linear complementarity problem will be established, which is a generalization of multisplitting parallel methods for solving the symmetric positive definite linear systems [5]. Some sufficient conditions for convergence of the multisplitting parallel methods will be proposed, when the matrix A is an H-matrix with positive diagonal elements positive definite.
We shall use the following notation. A real nn matrix A(aij) is called nonnegative (respectively, positive) and denoted A0 (respectively, A0) if aij 0 (respectively, aij0) for
n j
i
,
1 . In accordance with this notation, if A,B are real nn matrices, we say that AB
(respectively,AB) if AB0 (respectively,AB0) and similarly for vectors. A real matrix A
Definition 1.1([7]) A nonsingular matrix A aij Rn n
( ) is called an M -matrix if aij 0 for
j
i and A10. The comparison matrix A bij Rn n
( ) of a matrix A aij Rn n
( ) is defined
by i j n
j i a
j i a b
ij ij
ij , 1,2, , |,
| |, |
. A is said to be an H -matrix if its comparison matrix A is an
M -matrix.
Definition 1.2([2]) Let A be an nn matrix. The representation AMN is called a splitting of A if M is a nonsingular matrix. In addition, the splitting is (1) convergent if (M1N)1; (2)
regular if M10 and N 0.
Definition 1.3([9]) Let A be a nonsingular real nn matrix, and suppose that for some N, we are given matrices Mi,Ni,EiRnn,i1,2,,, satisfying
(1) AMiNi for i1,2,,. (2) Mi is nonsingular for i1,2,,.
(3) Ei is a diagonal matrix with nonnegative entries for i1,2,,, and E I
i i
1
(nn identity
matrix).
Then the collection of triples (Mi,Ni,Ei),i1,2,, is called a multisplitting of A.
Definition 1.4([1]) A matrix ARnn is said to be a P-matrix if all its principal minors are positive. The class of such matrices is denote P.
The Multisplitting Iterative Method for the Linear Complementarity Problem
In order to achieve higher computing efficiency by making use of the information contained in the matrix A, based on the matrix splitting iteration method, we present a multisplitting iterative method for solving the linear complementarity problem (1).
Let the matrix A is a symmetric positive definite matrix, AMi Ni(i1,2,,m) is the
multisplitting of the A,N{1,2,,n},Ni N(i1,2,m). Let
) , , 2 , 1 ( )
, , ,
( (1) (2) ( ) R i m
diag
Ei i i in n n
i
j
i 0,jN )
(
. (2) Then, we definite the multisplitting iterative method for the linear complementarity problem as follows:
Algorithm 2.1 (The multisplitting iterative method for the linear complementarity problem ) Step 1: Let z(0)be an arbitrary vector, 0, s(i,k),i1,2,,m and set k = 0.
Step 2: For each l{1,2,,s(i,k)}, let yi(L1) z(k1).
Step 3: Parallel Calculate {yi(l),i1,2,,m;l1,2,,s(i,k)}where yi(l) is the arbitrary solution:
. 0 ) (
) (
, 0 ,
0
) ( )
( )
(
) ( )
( ) (
l i i k i T
l i
l i i k i l i
y M z
N q y
y M z
N q
y
(3) Step 4: Calculate Eik ,i 1,2, ,m
)
(
, where
m
i
k i s i k i
T T
E
y E z
z q Az z
i
1
)) , ( ( ) (
) 2 (
min
Step 5: Then, set
m
i
k i s i k i k
y E z
1
)) , ( ( ) ( )
1 (
(5) If || Ax(k)q||2 hold, then stop. Otherwise, let k := k + 1 and return to Step 2.
Convergence Theorems
In this section, we establish the convergence theory for Method 3.1 when the system matrix A of the )
, (q A
LCP is an H -matrix with positive diagonal entries.
Lemma 3.1 [1] If ARnn is an H-matrix with positive diagonal elements positive definite, then the linear complementarity problem (1) has a unique solution for all vectors qRn.
Lemma 3.2 [1] If ARnn is positive definite, then the linear complementarity problem (1) has a unique solution for all vectors qRn.
Lemma 3.3 A matrix ARnn is a P-matrix if and only if the linear complementarity problem (1) has a unique solution for all vectors n
R
q .
Lemma 3.4 LetARnn be a positive definite matrix, and letAMN be a P-regular splitting of A. Then there exists r0 such that
1 ||
|| A1/2M1NA1/2 2r .
Lemma 3.5 Let AMi Ni(i 1,2,,m) a multisplitting of A, where A and Mi are H-matrix with positive diagonal elements respective. Let z* be a unique solution of the linear complementarity problem (1). Then, for an arbitrary vector qRn and any starting vector z0Rn, the uniquely defined sequence {zk}, generated by Algorithm 2.1, satisfies
| ||
| |
|y() z* N z z*
M i k
l i
i
. (6) Proof. To establish the expression (6), we first remark that yi(l) is uniquely defined because
) , , 2 , 1
(i m
Mi is an H-matrix with positive diagonal elements (see Lemma 3.1). We verify (6) component by component. Consider an arbitrary index j and assume that
j l
i j l
i z y z
y | ( )
| () * () *
Under this assumption, the inequality (6) holds clearly if ( ())j 0
l i
y , because the j-th component
of the left-hand vector in (6) is then nonpositive and the right-hand component is always nonnegative. Now, suppose that ( ())j 0
l i
y . Then, according to yi(l) yr,l1 be an arbitrary solution of the following linear complementarity problem (3), we have
0 )
( () ()
j l i i k
iz M y
N q
. On the other hand, we also have
0 )
( * ()
j l i i iz M y
N q
.
Subtracting the last two expressions and rearranging terms, we deduce
0 ) (
)
( ( ) () * () j
l i i i
j l i i k
iz M y q N z M y
N q
j k i j l i
i y z N z z
M ( )) ( ( ))
( () * ( ) * which implies | || | |
|y() z* N z z*
M i k
l i
i
(Because Miis an H-matrix with positive diagonal elements and|yi(l) z* |j(yi(l) z*)j)
In a similar fashion, we may establish the same inequality (6) if l j
i j
l
i z z y
y | ( )
| () * * () .
Consequently, the inequality (6) must hold.
Theorem 3.6 Let AMi Ni(i1,2,,m) a multisplitting of A , where A and Mi are H-matrix with positive diagonal elements respective. Let z* be a unique solution of the linear complementarity problem (1). Let Ei diag( i(1), i(2),, i(n))Rn n(i 1,2,,m)
. Where
i j
i 0,jN )
(
. If there exists
0
0 i
i N
M
A be a P-regular splitting of A and
0 i
N , Then, for
any initial vector z0, the iterative sequence {zk}, generated by the Algorithm 2.1 converges to the unique solution z* of the linear complementarity problem (1).
Proof. Let z* be the unique solution of the LCP(q,A), where the uniqueness comes from the assumption that A is an H -matrix with its diagonal elements. By the step 3 of the Algorithm 2.1 and Lemma 3.5, we have
| ||
| |
| yi(l)z* Mi 1 Ni zk z* .
By the expression (4) of the Algorithm 2.1, we have
) ( ) (
min z z* A z z*
i
E .
It follows from Lemma 3.5 that
2 * ) 0 ( 2 / 1 2 2 / 1 1 2 / 1 2 * ) 1 ( 2 / 1 2 2 / 1 1 2 / 1 2 * ) 1 ( 1 2 / 1 2 * ) ( 2 / 1 2 * ) ( 2 / 1 || ) ( || || ) ( || || ) ( || || ) ( || || ) )( ) (( || || ) ( || || ) ( || 0 0 0 0 0 0 0 z z A A N M A z z A A N M A z z N M A z z A z z A k i i k i i k i i k i k
We know from Lemma 3.4 that
1 || ) ( || 2 2 / 1 1 2 / 1 0
0
r A
N M
A i i .
Therefore 0 || ) ( ||
lim 1/2 ( ) * 2
A z z
k k . Thus 0 || ||
lim ( ) * 2
z z
k k
This completes the proof.
Acknowledgement
References
[1] R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, San Diego, 1992.
[2] Z.Z. Bai and T.Z. Huang, Accelerated overrelaxation methods for solving linear complementarity problem. J. UEST China, 1994, 23,428-432.
[3] B.-X. Duan, C.-L. Li, A.-N. Xu, Relaxed parallel multisplitting iterative algorithm for linear complementarity problems, OR Transactions, 2006, 10(3)77-84.
[4] N. Machida, M. Fukushima, T. Ibaraki, A multisplitting method for symmetric linear complementarity problems. J. Comput. Appl. Math., 1995, 62 217-227.
[5] Rui-ping Wen, Guo-yan Meng and Chuan-long Wang. Multisplitting iterative methods with general weighting matrices for solving symmetric positive linear systems. Mathematica Numerica Sinica, 2014, 36(1):27-34.
[6] O’Leary D P, White R. Multi-splitting of matrices and parallel solutions of linear systems. SIAM J. Alg. Discr, Meth., 1985,6:630-640.
[7] J.-L. Dong, M.-Q. Jiang, A modified modulus method for symmetric positive-definite linear complementarity problems. Numer. Linear Algebra 2009, Appl. 16, 129–143.
[8] A. Frommer and D. B. Szyld, H-splitting and two-stage iterative methods, Numer. Math., 1992,63, 345-356.
