I-10-AlES 936 NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE 1/11 OF COMPLENENTARY SO.. (U) GEORGIA INST OF TECH ATLANTAI PRODUCTION AND DISTRIBUTION RESE.. F A AL-KNAVYALIUNCLSSIFIED NOV 85 PDRC-9-05 N±4-3-K- 47 F012/INL

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NCESARANDUFICEICNDTIONS FOR TH KAR LSE

Q AND Q

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1Faiz A. A1-Kilayyal

PDRC 85-05

PROUCTONand

DISTRIBUTION RESEARCqHCENTER

SCHOOL OF INDUSTRIAL '.?

LIJ ANO SYSTEMSENGINEERINGGEORGIA INSTITUTE OF TECHNOLOGYA U TNIT OF THE UNIVERSITY SYSTEM.OF GEORGIA

C...,ATLANTA. GEORGIA 30332

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PDRC Report Series 85-05November 1985

401.

NECESSARY AND SUFFICIENT CONDITIONS FOR THEEXISTENCE OF COMPLEMENTARY SOLUTIONS ANDCHARACTERIZATIONS OF THE MATRIX CLASSES

Q AND O

by

Faiz A. Al-Khayyal

PDRC 85-05

School of Industrial and Systems Engineering

Georgia Institute of TechnologyAtlanta, Georgia 30332 K .-'

This work was supported by the Office of Naval Research under Contract No.N00014-83-K-0147 and by general research development funds provided by theGeorgia Institute of Technology. Reproduction is permitted in whole or inpart for any purpose of the U. S. Government.

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............................ . . . . . ..-.. .

Necessary and Sufficient Conditions for the Existence of ComplementarySolutions and Characterizations of the Matrix Classes Q and

"- Faiz A. Al-KhayyalSchool of Industrial and Systems Engineering

Georgia Institute of TechnologyAtlanta, Georgia 30332-0205

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provided by the Georgia Institute of Technology. Reproduction is .-.-~ ~permitted in whole or in part for any purpose of the U.S. Government. '-

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* Abstract

Necessary and Sufficient Conditions for the Existence of ComplementarySolutions and Characterizations of the Matrix Classes Q and QO

Faiz A. Al-KhayyalGeorgia Institute of Technology

The linear complementarity problem has a solution if and only if

there exists a right-hand-side vector of an associated dual linear pro-

gram such that at least one LP solution falls inside a specific open set.

This condition is used to give different characterizatons of the matrix

classes Q and Q0 " Moreover, they are used to derive verifiable suffi-

cient conditions for a subclass of Q.

-4"..

. Key Words: Linear Complementarity Problem, Q-Matrices, Q0-Matrices

Abbreviated Title: Conditions for LCP Solutions

This work was supported in part by the Office of Naval Research underContract No. N00014-83-K-0147 and by general research development fundsprovided by the Georgia Institute of Technology. Reproduction ispermitted in whole or in part for any purpose of the U.S. Government.

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Necessary and Sufficient Conditions for the Existence of Complementary

Solutions and Characterizations of the Matrix Classes Q and Q0

Faiz A. Ai-Khayyal .

Georgia Institute of Technology - v.

1. Introduction

The linear complementarity problem is to find a real n-vector x such

that

.5 T

.x+q 0, x, 0, x (',x+q) =0, (1)

where q is a known real n-vector and M is a known real nxn matrix. The

problem is generally denoted by the pair (q,M). For given M, the feasi-

ble region of (q,M) is given by the convex polyhedron -.

S(q,M) {xRn: .4x+q 0, x 01. (2)

When S(q,M) * 0, the problem (q,M) and the vector q are said to be feasi-

ble. A feasible problem (q,M) does not necessarily have a solution. The -

main theorem of this paper provides a necessary and sufficient condition

for problem (q,M) to have a solution. This result is then used to

characterize two important matrix classes.

Numerous procedures have been proposed for solving problem (q,M),

including Lemke [111, Cottle and Dantzig [3], Murty [14], Eaves [71,

mr-Saigal [18], Garcia [8], and Karamardian [9]. More recently, Mangasarian

[12,13], Cottle and Pang [4], Shiau [19], Pardalos and Rosen [16], ".

Ramarao and Shetty [171, and Al-Khayyal [11 have developed different .5%

approaches for solving either (I) or an equivalent restatement of the

2.2 - ,---.

2

problem. While the preceding list is not exhaustive, with the exception

of [1,16,171, all the methods assume structure in the matrix M. This

note addresses two such classes of structured matrices.

The classes of matrices considered are

Q'" {Rnn qR n => (q,M) has a solution)

and

Q0 {MeRnxn: qcRn and S(q,M) * 0 > (q,M) has a solution}.

The class Q0 has also been denoted K in the literature (e.g., Pang [15]).

Several authors have investigated these classes and have obtained full or

partial characterizations of such matrices (see, e.g., Mangasarian [121,

Cottle and Pang [41, Pang [151, Stone [20], Shiau [19], Doverspike and

Lemke [6], Kelly and Watson [101, and Cottle, Randow and Stone [5]). .i

A geometric characterization of the members of Q^ is that the union

of complementary cones of any matrix in Q0 is a convex set. The class Q

is a subset of O0 for which the union of complementary cones of any

member in Q is Rn. In this note, we obtain new necessary and sufficient

conditions for the classes Q and Q0 which have a different geometric

interpretation. Verifiable conditions based on linear programming dre

provided which are sufficient for membership in Q.

2. The Conditions

Let Y denote the unit hypercube

nY {ys R 0yce,

. . . . . .. . . . .. . . . . . . . . . . . . . . . .

.... .,.. ... ................... ...... ....-.......-.....- ,...... . -:":

., .. *x*-P V P '1 .? ~'.F.Prs ~ '. r~ ~ Tm T~ R in im S 'A nX- i M D -M '%A -n YWm -. '%-.W r- W" -. - U. -

3 1*.

Twhere e (,,.l) is the summation vector. For fixed ycY, define

the polyhedral set W I

V(y,M) =(vcR M v e-y, v ~-l

and consider the linear program

T

argmin~q v: vcV(y,M)),

and consider the following open polyhedral set

V-(v~e C-)v<e). (4)

n nxnTheorem 1. Let qcR7 and McR The problem (q,M) has a solution if and

only if there exists a vector ycY such that

T-argmin{q v: vcV(y,M)} n V * .(5)

C.. Proof: (Necessity). Let xsolve the problem (q,M). Let ycY be m

defined as follows

4(1 f

o if i o..'0 ifx 0

-.-. ,

It follows from linear programming duality that

0 e argmin(qTv: VEV(Y,M))n v

so that (5) is satisfied.

(Sufficiency). Let ycY be such that (5) holds and let

Ty v e argminfq v: vcV(y,M)} n V. Then by the Karush-Kuhn-

Tucker Conditions there exist multipliers (XE) with XcS(q,M)

and { = M-+q such that

T -X [e-y-Mtvi = 0

,-- (6)''

t[;+l ..o.&T [y v] = .

It follows that X T(MX+q) = XT = 0 so that X solves (q,M). 2-

For if this were not true, then there would exist an index i

such that Xi~ t = Xi(MX+q)i > 0. By the complementary slack-

ness condition (6), we must have

[e y MTv] =0 and [y + v]= 0.

Adding gives [e + (I-M) vi 0 which contradicts the assump-

-.. ,.

ew

*. . * * . . .. .. ... .. * .. . ... . . .,:;a:

5,.4

tion that v V.

Note that, when (5) holds for some ycY, problem (q,M) has a solution

and q is feasible. Thus, any infeasible q will violate (5) for all yCY.

The theorem can be restated to characterize the classes Q and Q0 "Corollary 1. Let qc n and nxn The matrix MeQ if and only if for

each q there exists a vector yeY such that (5) holds.

Corollary 2. Let qeR and McR" . The matrix McQ if and only if for

each feasible q there exists a vector ycY such that (5)

holds.

From the proof of the theorem, we see that when a complementary

solution exists, there exists a ycY such that the origin must necessarily

solve the linear program (3). The following example illustrates that

this need not be the case when q is feasible and no complementary

*solution exists.

Example 1. M [ 1 q [".

Here, we have S(q,M) = {x: 1/2xl+X2 +l 0 0, xl 1/2x2 -1 0 0,

x1 0 0, x2 0 0 , but there is no ycY for which the origin

solves problem (3). In fact, the optimal solution to the

linear program (3) is not in V for any yeY.

L% %Formalizing these observations, let cl V denote the closure of the

set V and let D(cl V) denote the boundary of the set cl V. We have the

°* . . • - . .•.. °= * . -° - • . .° °. .% . % * o . . -. - • .-. . -°.S, . .°-A. .' .'

. ~ ~ ~ ~ ~ % '% jyyj. yK Lr.' PF.P -. %'j -IF rw w -.. -11 pl. -W-. a I£ I- . %;-K7 -71. . .

6

following consequence of the theorem.

n nxnCorollary 3. Let qeRn and McR . The problem (q,M) does not have a

solution if and only if for every y c Y,*

argmin~q TV: vCV(y,M)) C 3(cl v.(7)

Proof: From Theorem 1, it follows that problem (q,M) does not have

a solution if and only if for every yY

argmin~q Tv: vcV(-y,M)} n V = .(8)

However, it is clear that

T-argmin{q v: vcV(y,M)} c cl V. (9)

p The result follows since (8) and (9) are true if and only

if (7) holds.

For completeness, this will be stated in terms of the classes Qand

Q0.

Corollary 4. Let qce and Me x. The matrix MeQ if and only if there

exists a q such that (7) holds for every ycY.

Corollary 5. Let qeR" and Mc The matrix MJQ 0 if and only if thereIN

%7-' : -7

7

exists a feasible q such that (7) holds for every ycY.

New sufficient conditions for membership in Qcan also be derived

from Theorem 1. Let

V {veR:M 4 e, v) -e). (10)

Corollary 6. If V' cVthen MeQ.

Proof: For every yeY, we have V(y,M) c V. Hence, for any qcR,

Tif V, C V then argminfq V: vcV(YM)) C V so that (5) is

satisfied. Thus every qeRn yields a solution to problem

(q,M) so that McQ.

3 The condition in Corollary 6 is rather strong since it implies that

(5) is satisfied for all ycY. Nevertheless, it includes some P-matrices

as illustrated by the following example.

Exaple2. M =[-1/2 12

V, {(v1,v2) 2v -v2 2, v +2v2 4 2, v1 > -1, v2 > -11

c{(Vlv1 v) v1 < 2, v2 > -2) =v.

On the other hanid, the next example shows that there are P-matrices

that violate the condition V cV.

8 .*....,'

8

Example 3. M = 1 ,

V" : {(vl,v 2 ): vl-4v2 2, vl+2v2 4 2, v I > -1, v2 ) -1},2" 1 v 2 1 2 1

V = {(vlv 2 ): vI + 4v > -2, v I < 2}.

* Here, V' V since (-i,-3/4) is in V' but not in V.

The preceding corollary suggests checkable conditions to test

whether or not V' c V. First, recall that a nonzero vector d is called a

direction of a set if the ray emanating from every point in the set in

the direction d is entirely contained in the set (see, e.g., Bazaraa and

Jarvis [2], Chapter 2). Let D(V') and D(V) denote the set of directions

of V' and V, respectively. These sets define the polyhedral cones

D(V') = {d * 0: MTd 0, d > 0}

OR' and

TDCV) { d *0: CM-I) d 0}.

Note that

D(V') _ D(V). Cli)

Theorem 2. The set V' is a subset of V if and only if for every extreme

point v of V',

. - ..

. .•-.

either (M v) =1 and v> 0ii

(12)

or <v <0 andy -

for all i =l2..n

Proof: (Necessity). Suppose V'c V and let v denote an arbitrary

extreme point of W'. Since v is an extreme point, then at

least n linearly independent constraints of the 2n

constraints of V' must be binding at v;thus,

either (MT-) 1 or vi=- (13)

for all i 1 ,2,...,n. But V'c V so that

I(-V) - < 1 (14)

for all i 1 ,2,...,n. Clearly, (13) and (14) imply (12).

(Sufficiency). Assume that the set V' has k extreme points

(v 'v ,...,v Iand Z. extreme directions (d 2d ...'dL). By

(12), we have

(>-I T < e (15)

for all j 1 ,2,...,k. Since V' is a convex polyhedral set,

then veV' if and only if there exist nonnegative vectors

(X,ii) with e TX 1 such that

k IrO .-.d,.

j=1 j-1 M Fe2

Hence,

k Lv -.-- '

7 = ( , TMvIE'Tvi + E U •1-"0T di< e.

The last inequality follows from (15) and (11) and the condi-

Ttions e X I and V 0. Therefore veV, which completes the

proof.

3. Summary and Conclusions

This note presents new necessary and sufficient conditions for the

existence of a solution to the linear complementarity problem. These

translate to similar conditions for the classes of matrices Q and Q and

for their complements. In particular, Theorem I (Corollary 2) states

that existence of a solution to the linear complementarity problem (mem-

bership in Q0) is equivalent to the existence of a solution to the linear

program (3), for some fixed ycY, which is also in the open set V given by

(4). From this follows the interesting condition on membership in the

complement of Q0, which is equivalent to the existence of a feasible q

such that every solution of the linear program (3), for every fixed yeY,

is on the boundary of the closure of V.

While these conditions are not easy to verify, they do provide a new

framework from which new results can be derived. For example, Corollary --.

6 provides a sufficient condition for membership in Q. This condition '

can be verified by using the test suggested by Theorem 2. This test

involves generating all the extreme points of V' and checking each U,"m ..

against (12). Note that to establish that a matrix is not in Q would

require generating all the extreme points only in the worst case. flow-

ever, to confirm that a matrix is in Q, all extreme points must be gene-

rated.

Acknowledgements. The author wishes to thank Richard E. Stone, Donna

.- Crystal, and John Vande Vate for their valuable comments on an earlier

dW draft of this paper.

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REFERENCES

[1] A1-Khayyal, F. A., "An implicit enumeration procedure for thegeneral linear complementarity problem," Industrial and SystemsEngineering Report Series No. J-83-7, Georgia Institute of -

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[2] Bazaraa, M. S., and Jarvis, J. J., Linear Programming and NetworkFlows, John Wiley & Sons, New York, 1977.

[3] Cottle, R. W., and Dantzig, G. B., "Complementary pivot theory ofmathematical programming," Linear Algebra and Its Applications 1(1968) 103-125.

[4] Cottle, R. W., and Pang, J. S., "On solving linear complementarityproblems as linear programs," Mathematical Programming Study 7(1978) 88-107.

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(6] Doverspike, R. D., and Lemke, C. E., "A partial characterization ofa class of matrices defined by solutions to the linearcomplementarity problem," Mathematics of Operations Research 7(1982) 272-294.

[7] Eaves, B. C., "The linear complementarity problem," ManagementScience 17 (1971) 612-634.

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[101 Kelly, L. M. and Watson, L. T., "Q-matrices and sphericalgeometry," Linear Algebra and Its Applications 25 (1979) 175-189.

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[141 Murty, K. G., "Note on a Bard-type scheme for solving the linearcomplementarity problem," Opsearch 11 (1974) 123-130.

,. .. . % , ", .-. , .,; j , .• . -. .".-* .•.. , •• . . .•. •. .-. . - ,,-. , .%'- .- . ' .. •, . % .. ,,,, .. -. . .. _

131

[151 Pang, J. S., "A note on an open problem in linear complementarity,"

Mathematical Programming 13 (1977) 360-363.

[16] Pardalos, P. M., and Rosen, J. B., "Global optimization approach tothe linear complementarity problem," Technical Report 84-37,Computer Sciences Department, University of Minnesota, Minneapolis,Minnesota, 1984.

[17] Ramarao, B., and Shetty, C. M., "Application of disjunctiveprogramming to the linear complementarity problem," Naval Research 'Logistics Quarterly 31 (1984) 589-600.

[18] Saigal, R., "On the class of complementary cones and Lemke'salgorithm," SIAM J. Appl. Math. 23 (1972) 46-50.

[19] Shiau, T. H., "Iterative linear programming for linearcomplementarity and related problems," Computer Sciences TechnicalReport No. 507, University of Wisconsin, Madison, Wisconsin, 1983.

(201 Stone, R. E., "Geometric aspects of the linear complementarityproblem," Ph.D. dissertation, Technical Report SOL 81-6, SystemsOptimization Laboratory, Stanford University, Stanford, California,1981.

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