A support submatrix for the generalized linear complementarity problem

Appl.

Pergamon

Lett. Vol. 7, No. 6, pp. 35-38, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved Math.

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A Support Submatrix for the Generalized Linear Complementarity Problem A. Department

EBIEFUNG

of Mathematics, University of Tennessee at Chattanooga Chattanooga, TN 37415, U.S.A. (Received May 1994; accepted June 1994)

Abstract-we provide an algorithm that selects, in a polynomial time, a representative submatrix whose appropriately defined LCP solution solves the GLCP. An algorithm based on support

submatrices is also presented. Keywords-Complementarity

problem, Algorithm, Support, Submatrix, Representative.

1. INTRODUCTION A vertical

block matrix

m x n, m > n, is said to be of type (ml,.

row-wise into n blocks so that the 3‘th block is of dimension

partitioned is called

N of dimension

a representative

. . , m,)

mj x n. A square matrix

of N if the jth row of M is from any row of the jth

submatrix

if N is M

block

of N. Given

a vertical

plementarity

block matrix

problem

of type (ml,.

[l] associated

. . , m,)

and q in IWm, the generalized

with N and q, denoted

GLCP(q,

N),

linear com-

is to find a vector

t

in Wn such that

9

2 (Nj

.z + q:)

= 0,

j=

l,...,n.

i=l

Different solution methods for problem GLCP(q, N) h ave been presented [l-6]. In this paper, we present an algorithm that selects a representative submatrix whose LCP solution is a solution of problem GLCP(q, N), i f one exists. This representative submatrix shall be referred to as a support submatrix for problem GLCP(q, N). An algorithm for solving problem GLCP(q, N) using support, submatrices is also demonstrated.

2. ALGORITHMS The first, algorithm selects a support submatrix for problem GLCP(q, N). We assume that problem GLCP(q, N) has a solution. The second algorithm solves problem GLCP(q, N) using support submatrices. The following theorem [2] is insightful in what follows. THEOREM 1. Let N be the vertical block matrix of type (ml, . . . , m,) and q a vector in Iw”. Problem GLCP(q, N) has a solution if and only if there exists a representative submatrix M and Research supported by UTC Faculty Research Grant.

Typeset by d&-m 35 AML 7:6-D

36

A. EBIEFUNG

a vector q in Wn, defined from q by taking entries corresponding to the rows of M, such that LCP(q, M) has a solution z that satisfies Nz + q 2 0. PROOF. Omitted. ALGORITHM 1. STEP 1. For each j, j = 1,. . . , n, solve the following LPs. fi(z) = min (Nj z + qj)i,

i=l,...,mj,

s.t. Nz + q L 0,

z > 0.

STEP 2. For each j, let ij, 1 2 ij 5 mj, be an index at which min(fi(z); Define a matrix M and a vector q in Rn by

Mj. =

(Nj)ij,

qj

7

i = 1,. . . , mj}

occurs.

=d.

3-

The solution of problem LCP(q, M) solves problem GLCP(q, N) as the next theorem shows. THEOREM 2. z is a solution of problem GLCP(q, N) if and only if it is a solution of problem LCP(q,

M).

PROOF. Suppose z is a solution of problem GLCP(q, N). Then z is feasible to problem LCP(q, M) by the way we defined M and q. Thus, w = Mz + q 2 0, z 2 0. This implies 0

wherei=l,...,rnj.


I (Njz+&

(1)

Hence 7 w$=Oif 9

j=l

“.i n(Nj.+qj)i=O,

,*.*,n*

i=l

Thus, z solves problem LCP(q, M). Conversely, if z solves LCP(q, M), then it is feasible to problem GLCP(q, N) by (1). By (1) and the complementarity condition 0 = ztw, we have that

zj l<~in_ --

((Nj

z)f + qj) = 0,

3

if and only if 1& --,

((Nzj):

+d)

= 0,

orzj=O,ifandonlyifforsomek,l
+ 4)

= 0,

0,

j=l

by nonnegativity conditions. Consequently,

Zj 3

(Nj

Z + qj)i

=

, * . . 7 n.

i=l

z solves problem GLCP(q, N). REMARK 1.

in a finite number of steps

is capable of selecting

A Support Submatrix

37

ALGORITHM 2. STEP 1. If q 2 0, then (W = q, z = 0) is a complementary solution.

Stop.

Otherwise go to

Step 2.

STEP 2. For each j, j = l,...

n, solve the LPs in Step 1 of Algorithm 1. If for some j there exists an i, 1 5 i 2 mj, such that the associated LP is infeasible or unbounded below, then problem GLCP(q, N) has no solution. Stop. Otherwise, select a matrix M and a vector q as in Step 2 of Algorithm 1.

STEP 3. Solve the LCP(q,

M) by appropriate method according to the class of M. If LCP(q,

has a solution (w, z), then (Nz + q, z) solves GLCP(q,N). solution. Stop.

Otherwise GLCP(q,N)

M)

has no

THEOREM 3. If for any j there exists an index ij, 1 5 ij 5 mj, such that the LP min Ni’, z + qij, s.t. Nz+q>O, is infeasible,

then problem

GLCP(q,

X20,

N) has no solution.

PROOF. Omitted. THEOREM 4. If problem

GLCP(q,

N) IS ’ f easible,

then the LPs

in Step 2 of the algorithm

bounded.

PROOF. Omitted.

-31

The validity of Algorithm 2 follows from Theorems 2, 3 and 4.

EXAMPLE 1. Let N and q be defined as follows.

N=

-1 -1 3 -2 -1

2-2 4, 6 3_

--1 q=

1 -1 -1

.

Suppose N is of type (2,3).

STEP 1. q is nonpositive. Go to Step 2. STEP 2. Solve the following LPs and select the associated matrices and vectors, j=l:

minzi + 222 - 1,

min - zi - 222 + 1,

s.t. Nz+qlO,

s.t. N z + q > 0,

~20.

z 2 0.

SOLUTIONS. Objective value: 0

0

Variables: 0

0

z2 = .5

.5

21

=

min{O, 0) = 0. This implies that

MI. = [1,2], j = 2:

qr = [-11,

min 3zi + 422 - 1, s.t. Nz + q 10,

ii& = [-1, -21,

min -2zi+6q-l, z 2 0.

s.t. Nz + q 2 0,

Qi = [l]. min zi + 322 - 1,

z 10.

s.t. Nz + q 2 0,

z 2 0.

are

A. EBIEFUNG

38 SOLUTIONS.

Objective value: 1

.3

Variables:

Zl

=

0

z2 = 05

.4

.4

.3

.3

min{ 1, 0,0.3} = 0. In this case, we have M2 = A&. = [-2,6],

q2 = 42 = [-11.

Thus,

M=-2[

1

2 6’

1

q=

4=

[_;I.

STEP 3. The algprithm selects two support submatrices that solve problem GLCP(q, IV), namely, M and M. Solving each LCP by appropriate method gives the z-vector as z = (.4, .3). The solution (Nz + q,z) = (O,O, 1.4,0, .3, .4, .3) solves problem GLCP(q, N).

REFERENCES 1. R.W. Cottle and G.B. Dantzig, A generalization of the linear complementarity problem, Journal of Combinatorial Theory 8, 79-90 (1970). 2. A.A. Ebiefung, The generalized linear complementarity problem and its application, Ph.D. Dissertation, Clemson University, Clemson, SC, (1991). 3. A.A. Ebiefung and M.M. Kostreva, Global solvability of generalized linear complementarity problems and a related class of polynomial complementarity problems, In Recent Advances in Global Optimization, (Edited by C. Floudas and P. Pardalos), Princeton University Press, Princeton, NJ, (1992). 4. O.L. Mangasarian, Generalized linear complementarity problems as linear programs, Operations Research Verjuhren 31, 393-402 (1979). of Mangasarian’s iterative algorithm for the generalized linear complementarity 5. M. Sun, Monotonicity problem, Journal of Mathematical Analysis and Applications 144, 474-485 (1989). 6. B.P. Szanc, The generalized complementarity problem, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY, (1989). 7. L. Vandenberghe, B.L. Demoor and J. Vanderwalle, The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits, IEEE lhnsaction on Circuits and Systems 11, 1382-1391 (1989).