ISBN 978-952-5726-02-2 (Print), 978-952-5726-03-9 (CD-ROM) Proceedings of the 2009 International Symposium on Information Processing (ISIP’09) Huangshan, P. R. China, August 21-23, 2009, pp. 316-318
Mixed Integer Linear Programming Method for Absolute Value Equations Longquan Yong Department of Mathematics, Shaanxi University of Technology, Hanzhong, China E-mail:
[email protected]
Abstract—We formulate the NP-hard absolute value equation as linear complementary problem when the singular values of A exceed one, and we proposed a mixed integer linear programming method to absolute value equation problem. The effectiveness of the method is demonstrated by its ability to solve random problems. Index Terms—absolute value equation; complementary problem; mixed integer programming.
I.
II.
Definition 2.1 The matrix Q is positive definite, i.e.,
d Qd > 0 for every 0 ≠ d ∈ R n . T
Definition 2.2 The linear complementarity problem [5] (LCP) is to determine a vector x ∈ R n satisfying xT (Mx + q) = 0, x ≥ 0, Mx + q ≥ 0 , (2)
linear linear
Where M ∈ R n×n and q ∈ R n .
INTRODUCTION
Proposition 2.3 Under the assumption that 1 is not an eigenvalue of A , the AVE (1) can be reduced to the following LCP (2): xT (Mx + q) = 0, x ≥ 0, Mx + q ≥ 0 ,
The NP-hard absolute value equation (AVE) :
Au − u = b
(1)
where A ∈ R n×n , u , b ∈ R n , and u denotes absolute value. A slightly more general form of the AVE was introduced in [1] and investigated in a more general context in [2]. The AVE (1) was investigated in detail theoretically in [1] and a bilinear program was prescribed there for the special case when the singular values of A are not less than one. No computational results were given in [1-2]. In contrast in [3], computational results were given for a linear-programming-based successive linearization algorithm utilizing a concave minimization model. A generalized Newton algorithm that is globally convergent under certain assumptions was introduced in context [4]. As was shown in [1], the general NP-hard linear complementarity problem (LCP) [2–4], which subsumes many mathematical programming problems, can be formulated as an AVE (1). This implies that (1) is NP-hard in its general form.
where
M = ( A + I )( A − I )−1 , q = (( A + I )( A − I )−1 − I )b .
Proof. According to reference [1], the AVE (1) is equivalent to the bilinear program
⎧0 = min ( ( A + I ) u − b )T ( ( A − I ) u − b ) ⎪ x∈R n ⎨ ⎪⎩ s.t. ( A + I ) u − b ≥ 0, ( A − I ) u − b ≥ 0
(3)
and the generalized LCP
0 ≤ ( ( A + I ) u − b ) ⊥ ( ( A − I ) u − b ) ≥ 0. Since 1 is not an eigenvalue of A , ( A − I )
−1
(4)
is existent.
Let x = ( A − I )u − b . Then
( A + I ) u − b = ( A + I ) ( A − I )−1 ( x + b) − b
The basic contribution of the present work is a simple algorithm based on a new reformulation of the AVE (1) as a mixed integer linear programming minimization problem. We prove that the solution to mixed integer linear programming is existent. This turns out to be an effective way for solving the AVE as indicated by computational results on some random AVEs.
= ( A + I )( A − I ) −1 x + (( A + I )( A − I ) −1 − I )b. So formula (4) can be reformulated following LCP:
xT (Mx + q) = 0, x ≥ 0, Mx + q ≥ 0 where
M = ( A + I )( A − I )−1 , q = (( A + I )( A − I )−1 − I )b .
In Sect. 2 of the present work we transform AVE into linear complementary problem when the singular values of A exceed 1. We solve linear complementary problem by mixed integer linear programming method in Sect. 3, and we prove that the solution to mixed integer linear programming is existent under certain assumptions. Effectiveness of the method is demonstrated in Sect. 4 by solving some randomly generated AVEs problem with singular values of A exceeding 1. Section 5 concludes the paper. Section 6 is acknowledgement.
© 2009 ACADEMY PUBLISHER AP-PROC-CS-09CN002
THE AVE PROBLEM AS LCP
Proposition 2.4 Assumed singular values of A exceeding 1. Then M = ( A + I )( A − I )−1 is positive definite. Proof. We first show that inverse of ( A − I ) exists. For, if not, then for some non-zero vector x ∈ R n we have that ( A − I ) x = 0 , which shows 1 is an eigenvalue of A , so 1 is singular values of A , too, 316
which gives the contradiction.
Theorem 3.1 The following mixed integer linear program
Because singular values of A exceeding 1, we get z T ( AT A − I ) z > 0 for every 0 ≠ z ∈ R n . Since
⎧max a ⎪s.t. 0 ≤ My + aq ≤ z, ⎪ ( MILP) ⎨ n ⎪ 0 ≤ y ≤ e − z, z ∈{0,1} , ⎪ 0 ≤ a ≤ 1, ⎩
z T ( AT A − I ) z > 0 ⇔ z T AT Az + z T AT z − z T Az − z T z > 0 ⇔ z T ( AT − I )( A + I ) z > 0,
associated to the LCP with q ≠ 0 has an optimal * * * * solution ( y , z , a ) satisfying a ≥ 0 , where
if we take z = ( A − I )−1 v ( z ≠ 0 ⇔ v ≠ 0) , then
e = (1,1,L,1) ∈Rn , y ∈ R n , z ∈ {0,1}n . T
z T ( AT − I )( A + I ) z > 0
⇔ vT ( AT − I )−1 ( AT − I )( A + I )( A − I )−1 v > 0
The LCP has a solution, and
⇔ v ( A + I )( A − I ) v > 0, (∀v ≠ 0), Hence M = ( A + I )( A − I )−1 is a positive definite −1
T
a* ≥ 0 . point ( y , z , a )
solution, if and only if Proof.
matrix.
The
a = yi = zi = 0
Lemma 2.5 The AVE (1) is uniquely solvable for any b ∈ R n if the singular values of A exceed 1. Proof. Since singular values of A exceed 1 it follows that inverse of ( A − I ) exists. By Proposition 2.3 we know that the AVE (1) can be reduced to the LCP (2). Combined Proposition 2.4, we get LCP (2) is monotone [6], Hence the LCP (2) is uniquely solvable for any q ∈ R n and so is the AVE (1)
( i = 1, 2,L , n )
is feasible for
defined
( MILP )
by
, and the
feasible domain of ( MILP ) is compact [12]. Therefore, *
*
*
an optimal solution ( y , z , a )
exists satisfying
a ≥ 0. *
a* ≥ 0 .
Let
for any b ∈ R n . III.
x* = y* / a* is a
0 ≤ a*xi* ≤1− zi*
and
x* = y* / a* satisfies
Then
0 ≤ a* ( Mx* + q ) ≤ zi* i
for
i =1,2,L,n . Since zi* ∈ {0,1} it follows that, for
SOLVING LCP BY MIXED INTEGER LINEAR PROGRAMMING
i = 1, 2,L , n ,
Linear complementarity problem is a fundamental problem in mathematical programming. It is known that any differentiable linear and quadratic programming can be formulated into a LCP. LCP also has wide range of applications in economic and engineering. The interested readers are referred to the survey paper [5].
( Mx
*
we
either
xi* = 0
have
or
+ q ) = 0 , and hence x = y / a is a solution *
*
*
i
to LCP. * Conversely, suppose that a = 0 . If the LCP had a solution x , then we cannot have x = 0 and Mx + q = 0,
A number of direct as well as iterative methods have been proposed for their solution. The book by Cottle et al. [5] is a good reference for pivoting methods developed to solve LCP. An other important class of methods used to tackle LCP are 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 linear programming [6]. 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 was first introduced in [6] and [7], Kojima et al. ([8]) proposed a polynomial time algorithm for monotone LCP under the nonemptiness assumption of the set of feasible interior point, and since then many other algorithms have been developed based on the primal-dual strategy [8-10].
because this would imply
q = 0 . By setting for
i = 1, 2,L , n ⎧⎪ 1, if ( Mx + q )i > 0 zi = ⎨ ⎪⎩0， if ( Mx + q )i = 0 and
a1 = min {1/ xi：xi > 0} , i
a2 = min {1/( Mx + q)i ： ( Mx + q )i > 0} , i
a = min {a1 , a2 ,1} , y = ax, we see that
( y, z, a )
would be a feasible solution of
( MILP ) . This contradicts ( MILP ) .
Following we indicate the LCP can also be formulated as a mixed integer linear program [11].
q = 0 (q ≥ 0) is of little interest since it always has the solution x = 0 . Notice that an LCP with
317
IV.
V. CONCLUSION
COMPUTATIONAL RESULTS
In this paper, we presented a new method for solving AVE problems when the singular values of A exceed 1. Preliminary numerical examples indicate that the proposed algorithm seems promising for solving AVE problems.
In this section we present a couple of numerical examples. We consider an instance of the AVE problem where the data (A, b) are generated by the Matlab scripts: rand('state',0); R=rand(n,n); b=rand(n,1); A=R'*R+n*eye(n); qiyizhi=svd(A); M=(A+eye(n))*(inv(A-eye(n))); q=((A+eye(n))*(inv(A-eye(n)))-eye(n))*b;
VI.
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) and the Foundation of Shaanxi University of Technology (No.SLGQD0517).
and we set the random-number generator to the state of 0 so that the same data can be regenerated. If we take n = 4 , then we obtain following AVE problem where
⎡5.5606 ⎢1.3090 A=⎢ ⎢1.6416 ⎢ ⎣1.3506
1.3090 1.6416 5.5839 1.3666 1.3666 5.8784 1.4722 1.5152 ⎡ 0.93547 ⎤ ⎢ 0.91690 ⎥ ⎥. b=⎢ ⎢ 0.41027 ⎥ ⎢ ⎥ ⎣ 0.89365 ⎦
REFERENCES
1.3506 ⎤ 1.4722 ⎥⎥ , 1.5152 ⎥ ⎥ 5.5904 ⎦
[1]
[2]
[3] [4]
[5]
We transform this problem into monotone linear complementarity problem and solve by mixed integer linear program. Using LINDO software, we get the following solution to ( MILP ) :
[6]
[7]
a = 1.000000, y = ( 0, 0,0.052639, 0 ) , ∗
Whereas
∗
T
[8]
a* = 1.000000 > 0 , thus T x* = y ∗ a∗ = ( 0, 0,0.052639, 0 )
is solution to corresponding LCP. solution to this AVE problem is
ACKNOWLEDGMENTS
[9]
Hence uniquely
u = ( A− I )−1(x + b) = ( 0.1420,0.1287. − 0.0263,0.1203)
[10]
.And finally, as can be seen from above, it is quite an easy task to implement the method. We generated random problems of dimension 30 to over 100 using the proposed method, and the experiments presented were performed on a personal compute. In all instances this method perform extremely well, and obtain its optimal solution respectively.
[11]
T
[12]
318
Mangasarian, O.L.,Meyer, R.R. Absolute value equations[J]. Linear Algebra and its Applications, 2006, 419(2): 359-367. Mangasarian, O.L. Absolute value programming [J]. Computational Optimization and Aplications, 2006,36(1): 43-53. Mangasarian, O.L. Absolute value equation solution via concave minimization [J]. Optim. Lett. 2007,1(1): 3-8. Mangasarian, O.L. A generlaized newton method for absolute value equations [J]. Optim. Lett. 2009, 3(1): 101108. R.W. Cottle, J.S Pang, and R E Stone, The Linear Complementarity Problems, Academic Press, 1992. Kojima M, Megiddo N, Yoshise A. A unified approach to interior point algorithms for linear complementary problem[C]. Lecture Notes in computer science 538. Berlin: Springer-Verlag.1991. M. Kojima, N. Megiddo, S. Mizuno, A Primal-dual Infeasible Interior Point Algorithm for Linear Programming, Math. Prog. 1993,61:261-280. M. Kojima, S. Mizuno, A. Yoshise, A Polynomial-time Algorithm for a Class of Linear Complementarity Problems, Math. Prog. 1989, 44:1-26. Y. Zhang, On the Convergence of a Class of InfeasibleInterior-Point Methods for the Horizontal Linear Complementarity Problem, SIMA J. Optim. 1994, 4:208227. S.J. Wright, An Infeasible-Interior-Point Algorithm for Linear Complementarily Problems, Math. Prog. 1994,67:29-52. Horst, R., Pardalos, P.M. Handbook of global optimization[M], Kluwer Academic Publishers, Dordrecht, 1994. BANSAL,P.P.,and JACOBSEN,S.E.,Characterization of Basic Solutions for a Clsss of Nonconvex Programs,Journal of Optimization Theory and Applications,1975,15: 549-564
