Abstract
The class of realn × n matricesM, known asK-matrices, for which the linear complementarity problemw − Mz = q, w ≥ 0, z ≥ 0, w T z =0 has a solution wheneverw − Mz =q, w ≥ 0, z ≥ 0 has a solution is characterized for dimensionsn <4. The characterization is finite and ‘practical’. Several necessary conditions, sufficient conditions, and counterexamples pertaining toK-matrices are also given. A finite characterization of completelyK-matrices (K-matrices all of whose principal submatrices are alsoK-matrices) is proved for dimensions <4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Aganagic and R.W. Cottle, “InQ-matrices”, Technical Report, Department of OR, Stanford University (Stanford, CA, 1979).
W.R. Ballard,Geometry (W.B. Saunders Company, Philadelphia, 1970).
R.W. Cottle, “Completely-Q-matrices“,Mathematical Programming 19 (1980) 347–351.
R.D. Doverspike, “Some perturbation results for the linear complementarity problem“,Mathematical Programming 23 (1982) 181–192.
R.D. Doverspike and C.E. Lemke, “A partial characterization of a class of matrices defined by solutions to the linear complementarity problem“,Mathematics of Operations Research 7 (1982) 272–294.
B.C. Eaves, “The linear complementarity problem“,Management Science 17 (1971) 612–634.
M. Fiedler and V. Pták, “On matrices with non-positive off-diagonal elements and positive principal minors“,Czechoslovak Mathematical Journal 12 (1962) 382–400.
C.B. Garcia, “Some classes of matrices in linear complementarity theory“,Mathematical Programming 5 (1973) 299–310.
M.J. Greenberg,Euclidean and non-euclidean geometries (Freeman, San Francisco, 1980).
A.W. Ingleton, “A problem in linear inequalities“,Proceedings of the London Mathematical Society 3rd Series 16 (1966) 519–536.
L.M. Kelly and L.T. Watson, “Q-matrices and spherical geometry“,Linear Algebra and its Applications 25 (1979) 175–189.
C.E. Lemke, “Bimatrix equilibrium points and mathematical programming“,Management Science 11 (1965) 681–689.
C.E. Lemke, “On complementary pivot theory“, in: G.B. Dantzig and A.F. Veinott, Jr., eds.,Mathematics of the Decision Sciences Part 1 (American Mathematical Society, Providence, RI, 1968) 95–114.
C.E. Lemke, “Recent results on complementarity problems“ in: J.B. Rosen, O.L. Mangasarian and K. Ritter, eds.,Nonlinear programming (Academic Press, New York, 1970) 349–384.
K.G. Murty, “On a characterization ofP-matrices“,SIAM Journal of Applied Mathematics 20(3) (1971) 378–384.
K.G. Murty, “On the number of solutions to the complementarity problem and spanning properties of complementary cones“,Linear Algebra and its Applications 5 (1972) 65–108.
H. Samelson, R.M. Thrall and O. Wesler, “A partitioning theorem for EuclideanN-space“,Proceedings of the American Mathematical Society 9 (1958) 805–807.
A. Tamir, “On a characterization ofP-matrices“,Mathematical Programming 4 (1973) 110–112.
L.T. Watson, “A variational approach to the linear complementarity problem”, Doctoral Dissertation, Department of Mathematics, University of Michigan (Ann Abor, MI, 1974).
Author information
Authors and Affiliations
Additional information
Partially supported by NSF Grant MCS-8207217.
Research partially supported by NSF Grant No. ECS-8401081.
Rights and permissions
About this article
Cite this article
Fredricksen, J.T., Watson, L.T. & Murty, K.G. A finite characterization ofK-matrices in dimensions less than four. Mathematical Programming 35, 17–31 (1986). https://doi.org/10.1007/BF01589438
Issue Date:
DOI: https://doi.org/10.1007/BF01589438