P e r g a m o n Appl. Math. Lett. Vol. 8, No. 1, pp. 97-100, 1995

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R e d u c i n g a M o n o t o n e H o r i z o n t a l L C P t o a n L C P

M. S. GOWDA Department of Mathematics and Statistics University of Maryland Baltimore County

Baltimore, MD 21228, U.S.A. gowda©math, umbc. edu

(Received and accepted February 1993)

A b s t r a c t - - I n this note, we show how to reduce a monotone horizontal linear complementarity problem to a (standard) monotone linear complementarity problem. The main steps involved in this reduction are: finding a maximal linearly independent column set of a given matrix (or equivalently, converting a matrix to its reduced row echelon form) and inverting a nonsingular matrix.

Keywords- -Hor izonta l LCP, Vertical LCP, Standard LCP, Monotone, Column representative.

1. I N T R O D U C T I O N

For a given matr ix M E N ~x~ and a vector q E R ~, the (standard) linear complementar i ty problem, LCP(M,q) , is to find a vector x E R ~ such tha t

x > O, y = M x + q > O, and xTy ---- 0. (1)

The importance of this problem is well documented in the literature; see for example, [1,2].

Currently, an important generalization of this problem called the horizontal linear complemen- tar i ty problem (HLCP) is being studied by several authors. Given two matrices A and B in R nxn and a vector q E R ~, HLCP(A, B, q) is to find vectors x and y in ] ~ such tha t

A y - B x = q, x >_ O, y >_ O, xTy = O. (2)

Clearly, when A is invertible, the above HLCP can be written as L C P ( A - 1 B , A - l q ) .

We shall say tha t the pair {A, B} (as given in (2)) has the column monotonici ty proper ty and call HLCP(A, B, q) a monotone HLCP if

A y - B x = 0 ~ x T y >__ O. (3)

For the monotone HLCP, interior point methods have been described by Zhang [3], Monteiro and Tsuchiya [4], and Billups and Ferris [5]; error bound results were studied by Luo and Pang [6]. For a s tudy of the column monotonicity property in electrical networks, see [7].

I t was observed by Sznajder and Gowda [8] that when the pair {A, B} has the column monotonicity property, HLCP(A, B, q) can be reduced to a standard monotone LCP. Their proof consists in showing tha t A + B is nonsingular and hence {A, B} has a nonsingular column rep- resentative which could then be used to reduce the given HLCP to a s tandard LCP. The actual

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98 M.S. GOWDA

method of finding such a nonsingular column representative (say, in polynomial time) was not discussed in the Sznajder and Gowda paper.

Independently, in [9], Tiitiincii and Todd describe a (graph-theoretic) algorithm of polynomial complexity. This algorithm starts with (any) given HLCP(A, B, q) and at the end produces an equivalent HLCP(A', B' , q') where either

(i) A' is the n × n identity matrix (in which case we have an LCP) or (ii)

A' = I and B' = * . 0 0

The algorithm works by successively finding matrices A (1), A (2) . . . . , A (p) where .4 (1) is the row reduced form with identity I in the north-west corner, A (p) = A', and rank A (k) increases strictly as k goes from 1 to p. Tiitiincii and Todd then prove that (ii) cannot hold when {.4, B} has the column monotonicity property (more generally when {A, B} is a P0-pair; see [9] for details).

The purpose of this note is to describe a simple way of rewriting (2), when {A, B } satisfies the column monotonicity property, as

Cv - Du = q, u > O, v >_ O, uTv = O, (4)

with C invertible and C - 1 D monotone (i.e., positive semidefinite), thereby reducing the given HLCP to the monotone L C P ( C - I D , C-lq).

2. T H E R E D U C T I O N

To see how (2) can be reduced to (4) with C - 1 D monotone, assume that {A,B} in (2) has the column monotonicity property. If A is nonsingular, we already have the monotone LCP(A-1B, A - l q ) . So assume that A is singular. With the observation that A ~ 0, we find a maximal set of linearly independent columns of A. (Note that this set can be identified in O(n 3) operations. For example, if A is reduced to its row echelon form, then the columns corresponding to the appropriate unit vectors will give such a maximal set.) Let {Ail, Ai2 , . - . , AiL} denote this maximal set. Using subscripts to denote columns, we define matrices C and D by

Aj if j C { i l , i2 , . . . , iL} and n j : ~ Bj if j E {i l , i2, . . . ,iL} cj (5) [ - B j otherwise [ - A j otherwise.

Correspondingly, we define new variables v and u by

{ xj i f j E { i l , i 2 , . . . , iL} yj i f j E { i l , i 2 , . . . ,iL} and uj v j = = (6 )

xj otherwise yj otherwise.

Clearly, (2) is now written as (4); furthermore, {C, D} has the column monotonicity property. We now show that C is nonsingular. For notational simplicity, we assume t h a t {il, i 2 , . . . , iL} = {1, 2 , . . . , L}. Suppose that C is singular, so that for some nonzero vector z C R n,

zlA1 + z2A2 . . . + ZLAL -- ZL+IBL+I . . . . znBn = O.

Since z ¢ 0 and the set {A1, A2 , . . . , AL} is linearly independent, we see that for some j > L + 1, zj ~ O. Without loss of generality, let ZL+l ~ O. Since the set {A1,A2, . . . ,AL+I} is linearly dependent, for some scalars Wl, w2, • •. , wL, we have

wlA1 + w2A2 + . . . + WLAL + AL+I = O.

Monotone Horizontal LCP 99

I t follows, from the above two equations, tha t for any scalar A,

(zl + Awl)A1 + (z: + A w 2 ) A 2 • • • + (ZL ÷ AWL)AL ÷ AAL+I - ZL+IBL+I . . . . z n B n = O.

Now the column monotonicity of {A, B} implies that )~ZL+I ~_ O. Since A is arbitrary, we must have z i + l = 0 leading to a contradiction. Hence C is nonsingular.

Clearly, the column monotonicity of {C, D} is equivalent to tha t of {I, C - 1 D } proving the monotonicity (i.e., positive semidefiniteness) of C - 1 D . Now multiplying the first equation in (4) by C -1, we see tha t (4) is equivalent to L C P ( C - 1 D , C - l q ) .

We end this note with two remarks.

REMARK 1. In the above argument, we picked an arbi trary maximal linearly independent set of columns in A and constructed the nonsingular matrix C. This breaks down if monotonicity is

not assumed. For example, let

A = I10 10] and B = [01 00].

Then the matr ix formed by the first column of A and the second column of B is singular. We also note tha t the pair {A, B} is a P0-pair [9], i.e., whenever A y - B x = 0 and (x, y) 7t (0, 0),

there exists an index i such that either xi or Yi is nonzero, and x iy i >_ O.

REMARK 2. Suppose tha t A, B E ~nxn, and a, b E R n. Consider the vertical linear complemen- tar i ty problem of finding a vector x E ]R n such that

Ax + a >_ O, B x + b > 0 , (Az ÷ a)T(Bx ÷ b) = O.

If { A T, B T} has the column monotonicity property (in which case we say tha t {A, B} has the row mono ton ic i t y property), our previous analysis produces two row representatives C and D of {A,B} (which means that for each index i, either C i = A i and D i = B i, or C i = B i and

D i -- A i where the superscript denotes the corresponding row) such tha t C is invertible and D C - 1 is positive semidefinite. The above problem can now be written as

C x ÷ c >_ O, D x ÷ d > O, ( C x ÷ c ) T ( D x ÷ d) --- 0,

which, upon putt ing z = C x + c, can be transformed into an LCP with the positive semidefinite matr ix D C -1.

R E F E R E N C E S

1. K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Heldermann Verlag, West Berlin, (1988).

2. R.W. Cottle, J.-S. Pang and R.E. Stone, The Linear Complementarity Problem, Academic Press, New York, (1992).

3. Y. Zhang, On the convergence of a class of infeasible interior-point algorithm for the horizontal linear complementarity problem, SIAM Journal on Optimization (1994) (to appear).

4. R.D.C. Monteiro and T. Tsuchiya, Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem, Research Report, Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ, (December 1992).

5. S.C. Billups and M.C. Ferris, Convergence of infeasible interior-point algorithms from arbitrary starting points, Computer Sciences Technical Report #1180, Computer Sciences Department, University of Wiscon- sin, Madison, WI, (October 1993).

6. Z.-Q. Luo and J.-S. Pang, Error bounds for analytic systems and their applications, Research Report, Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada, (January 1993).

7. A.N. Willson, Jr., Editor, Nonlinear Networks: Theory and Analysis, IEEE Press, New York, (1974). 8. R. Sznajder and M.S. Gowda, Generalizations of P0- and P-properties; extended vertical and horizontal

LCPs, Linear Algebra and Its Applications (1994) (to appear).

I00 M.S. GOWDA

9. R.H. Tiitiincii and M.J. Todd, Reducing horizontal linear complementarity problems, Research Report, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, (September 1993).

10. M.S. Gowda and R. Sznajder, The generalized order linear complementarity problem, SIAM Journal on Matrix Analysis and Applications (1994) (to appear).

11. O. Giiler, Generalized linear complementarity problems and interior point algorithms for their solutions, Technical Report, Faculty of Technical Mathematics and Computer Science, Delft University of Technology, The Netherlands, (April 1992; Revised September 1993).

12. A.N. Willson, Jr., A useful generalization of the Po-matrix concept, Numer. Math. 17, 62-70 (1971).