JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 21, No. 4, APRIL 1977

T E C H N I C A L N O T E

A Note on the Complementarity Problem 1

C. B. GARCIA 2

Communicated by S. Karamardian

Abstract. We show by an example that, in a complementarity problem where the given map is continuous and monotone" on the nonnegative orthant, the existence of a feasible solution is not sufficient to guarantee existence of a solution to the complementarity problem.

Key Words. Complementarity, mathematical programming, mono- tone maps, nonlinear programming.

1. Discussion

For a given map F: E " ~ E", the complementar i ty problem is to find ~ E " such that

----- 0, F(~) -> 0, i F ( i ) = 0.

It has been shown in Refs. 1-2 that, if F is continuous and monotone on E~, i.e.,

(x -y ) (F(x) -F(y) )~O

for all x, y E E l , and there exists an x - 0 , with F ( x ) > 0 , then the com- plementar i ty p rob lem has a solution. The question arises whether the continuity and monotonici ty of F and the existence of an x >- 0 with F(x) >- 0 is sufficient for the existence of a solution. The answer to this question is in the negative as the following example indicates.

1 T h e a u t h o r t h a n k s P r o f e s s o r S. K a r a m a r d i a n a n d D r . J . M o r e fo r he lp fu l d i scuss ions r e g a r d i n g th i s no te .

2 A s s i s t a n t P r o f e s s o r of M a n a g e m e n t Sc ience , G r a d u a t e S c h o o l o f Bus ines s , T h e U n i v e r s i t y of

C h i c a g o , C h i c a g o , I l l inois .

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530 JOTA: VOL. 21, NO. 4, APRIL 1977

Example 1.1. Let F : E 2 ~ E 2 be given by

FI(x) = - ( x 2 - 1 ) 2, F2(x) = 1 +2(x2-1)Xl.

Obviously, F is continuous on E2+. Also, F is monotone on E2+, since its Jacobian

[ 0 - 2 ( x 2 - 1 ) ] JF = 2(x2-1) 2xl .I

is positive semidefinite on E~+; see Ref. 3. At the point x = (0, 1), we have

Fa(0, 1) = 0, F2(0, 1) = 1.

However, for any x -> 0, and F(x) >- O, we have

Thus

x2 = 1, FI(X) = 0, and Fz(x) = 1.

imply that

x>--.O, F(x)>_O

xF(x)>O,

and the complementarity problem has no solution.

References

1. MORE, J., Classes of Functions and Feasibility Conditions in Nonlinear Com- plementarity Problems, Mathematical Programming, Vol. 6, pp. 327-338, 1974.

2. KARAMARDIAN, S., Complementarity Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, No. 4, 1976.

3. KARAMARDIAN, S., The Nonlinear Complementarity Problem with Applications, Part 1, Journal of Optimization Theory and Applications, Vol. 4, No. 2, 1969.