Abstract
The linear complementarity problem is the problem of finding solutionsw, z tow = q + Mz, w≥0,z≥0, andw T z=0, whereq is ann-dimensional constant column, andM is a given square matrix of dimensionn. In this paper, the author introduces a class of matrices such that for anyM in this class a solution to the above problem exists for all feasibleq, and such that Lemke's algorithm will yield a solution or demonstrate infeasibility. This class is a refinement of that introduced and characterized by Eaves. It is also shown that for someM in this class, there is an even number of solutions for all nondegenerateq, and that matrices for general quadratic programs and matrices for polymatrix games nicely relate to these matrices.
Similar content being viewed by others
References
R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming”,Linear Algebra and its Applications 1 (1968) 103–125.
B.C. Eaves, “The linear complementarity problem”,Management Science 17 (9) (1971) 612–634.
J.T. Howson, Jr., “Equilibria of polymatrix games”,Management Science 18 (5) (1972) 312–318.
S. Karamardian, “The complementarity problem”,Mathematical Programming 2 (1) (1972) 107–129.
C.E. Lemke, “On complementary pivot theory”, in:Mathematics of the decision sciences Eds. G.B. Dantzig and A.F. Veinott, Jr. (American Mathematical Society, Providence, R.I., 1968).
C.E. Lemke, “Recent results on complementarity theory”, in:Nonlinear programming (Academic Press, New York, 1970).
K.G. Murty, “On a characterization ofP-matrices”, Technical Report 69-20, Department of Industry and Engineering, University of Michigan, Ann Arbor, Mich. (May 1969).
R. Saigal, “A characterization of the constant parity property of the number of solutions to the linear complementarity problem”,SIAM Journal of Applied Mathematics 23 (1) (1972) 40–45.
H. Scarf and T. Hansen, “On the applications of a recent combinatorial algorithm”, Cowles Commision Discussion Paper No. 272 (April 1969).
Author information
Authors and Affiliations
Additional information
Research partially supported by National Science Foundation Grant NSF-GP-15031.
Rights and permissions
About this article
Cite this article
Garcia, C.B. Some classes of matrices in linear complementarity theory. Mathematical Programming 5, 299–310 (1973). https://doi.org/10.1007/BF01580135
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580135