Abstract
In this paper we consider the problem of establishing the number of solutions to the complementarity problem. For the case when the Jacobian of the mapping has all principal minors negative, and satisfies a condition at infinity, we prove that the problem has either 0, 1, 2 or 3 solutions. We also show that when the Jacobian has all principal minors positive, and satisfies a condition at infinity, the problem has a unique solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.W. Cottle, “Nonlinear programs with positively bounded Jacobians”,SIAM Journal on Applied Mathematics 14 (1966) 147–158.
D. Gale and H. Nikaido, “The Jacobian matrix and global univalence of mappings”,Mathematische Annalen 159 (1965) 81–93.
K. Inada, “Factor intensity and Stolper—Samuelson condition”, mimeographed (1966).
S. Karamardian, “The complementarity problem”,Mathematical Programming 2 (1972) 107–129.
M. Kojima, “A unification of existence theorems of the nonlinear complementarity problem”,Mathematical Programming 9 (1975) 257–277.
M. Kojima, “Studies on piecewise linear approximations of piecewise-C1 mappings in fixed points and complementarity theory”,Mathematics of Operational Research 3 (1978) 17–36.
M. Kojima and R. Saigal, “A study of PC1 homeomorphisms on subdivided polyhedrons”,SIAM Journal on Mathematical Analysis 10 (1979) 1299–1312.
M. Kojima and R. Saigal, “On the number of solutions for a class of linear complementarity problems”,Mathematical Programming 17 (1979) 136–139.
O.L. Mangasarian, “Locally unique solutions of quadratic programs, linear and nonlinear complementarity problems”, Computer Sciences Technical Report No. 375, Univ. of Wisconsin, Madison, WI (February 1979).
A. Mas-Colell, “Homeomorphisms of compact convex sets and the Jacobian matrix”,SIAM Journal on Mathematical Analysis 10 (1979) 1105–1109.
N. Megiddo and M. Kojima, “On the existence and uniqueness of solutions in the nonlinear complementarity theory”,Mathematical Programming 12 (1977) 110–130.
K.G. Murty, “On the number of solutions to the complementarity problem and the spanning properties of complementary cones”,Linear Algebra and its Applications 5 (1972) 65–108.
H. Nikaido,Convex structures and economic theory (Academic Press, New York, 1968).
J.M. Ortega and W.C. Rheinboldt,Iterative solutions of nonlinear equations in several variables (Academic Press, New York, 1970).
W.C. Rheinboldt and J.S. Vandergraft, “On piecewise affine mappings inR n”,SIAM Journal on Applied Mathematics 29 (1975) 680–689.
R. Saigal and C.B. Simon, “Generic properties of the complementarity problem”,Mathematical Programming 4 (1973) 324–335.
H. Samelson, R.M. Thrall and O. Wesler, “A partition theorem for Euclideann-space”,Proceedings of the American Mathematical Society 9 (1958) 805–807.
Author information
Authors and Affiliations
Additional information
Sponsored by the United States Army under Contract No. DAAG29-75-C-0024. This material is based upon work supported by the National Science Foundation under Grant No. MCS77-03472 and Grant No. MCS78-09525. This work appeared as an MRC Technical Report No. 1964, University of Wisconsin, Madison, WI, June 1979.
Rights and permissions
About this article
Cite this article
Kojima, M., Saigal, R. On the number of solutions to a class of complementarity problems. Mathematical Programming 21, 190–203 (1981). https://doi.org/10.1007/BF01584240
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01584240