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A finite characterization of K-matrices in dimensions less than four.(English)Zbl 0641.90082

The class of real \(n\times n\) matrices M, known as K-matrices, for which the linear complementarity problem \(w-Mz=q\), \(w\geq 0\), \(z\geq 0\), \(w^ Tz=0\) has a solution whenever \(w-Mz=q\), \(w\geq 0\), \(z\geq 0\) has a solution is characterized for dimensions \(n<4\). The characterization is finite and ‘practical’. Several necessary conditions, sufficient conditions, and counterexamples pertaining to K-matrices are also given. A finite characterization of completely K-matrices (K-matrices all of whose principal submatrices are also K-matrices) is proved for dimensions \(<4\).

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Cite

References:

[1]M. Aganagic and R.W. Cottle, ”InQ-matrices”, Technical Report, Department of OR, Stanford University (Stanford, CA, 1979). ·Zbl 0416.90074
[2]W.R. Ballard,Geometry (W.B. Saunders Company, Philadelphia, 1970).
[3]R.W. Cottle, ”Completely-Q-matrices”,Mathematical Programming 19 (1980) 347–351. ·Zbl 0442.90091 ·doi:10.1007/BF01581653
[4]R.D. Doverspike, ”Some perturbation results for the linear complementarity problem”,Mathematical Programming 23 (1982) 181–192. ·Zbl 0484.90087 ·doi:10.1007/BF01583787
[5]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. ·Zbl 0498.90077 ·doi:10.1287/moor.7.2.272
[6]B.C. Eaves, ”The linear complementarity problem”,Management Science 17 (1971) 612–634. ·Zbl 0228.15004 ·doi:10.1287/mnsc.17.9.612
[7]M. Fiedler and V. Pták, ”On matrices with non-positive off-diagonal elements and positive principal minors”,Czechoslovak Mathematical Journal 12 (1962) 382–400. ·Zbl 0131.24806
[8]C.B. Garcia, ”Some classes of matrices in linear complementarity theory”,Mathematical Programming 5 (1973) 299–310. ·Zbl 0284.90048 ·doi:10.1007/BF01580135
[9]M.J. Greenberg,Euclidean and non-euclidean geometries (Freeman, San Francisco, 1980). ·Zbl 0418.51001
[10]A.W. Ingleton, ”A problem in linear inequalities”,Proceedings of the London Mathematical Society 3rd Series 16 (1966) 519–536. ·Zbl 0166.03005 ·doi:10.1112/plms/s3-16.1.519
[11]L.M. Kelly and L.T. Watson, ”Q-matrices and spherical geometry”,Linear Algebra and its Applications 25 (1979) 175–189. ·Zbl 0421.90071 ·doi:10.1016/0024-3795(79)90017-X
[12]C.E. Lemke, ”Bimatrix equilibrium points and mathematical programming”,Management Science 11 (1965) 681–689. ·Zbl 0139.13103 ·doi:10.1287/mnsc.11.7.681
[13]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. ·Zbl 0208.45502
[14]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. ·Zbl 0227.90043
[15]K.G. Murty, ”On a characterization ofP-matrices”,SIAM Journal of Applied Mathematics 20(3) (1971) 378–384. ·Zbl 0224.90034 ·doi:10.1137/0120041
[16]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. ·Zbl 0241.90046 ·doi:10.1016/0024-3795(72)90019-5
[17]H. Samelson, R.M. Thrall and O. Wesler, ”A partitioning theorem for EuclideanN-space”,Proceedings of the American Mathematical Society 9 (1958) 805–807. ·Zbl 0117.37901
[18]A. Tamir, ”On a characterization ofP-matrices”,Mathematical Programming 4 (1973) 110–112. ·Zbl 0277.15008 ·doi:10.1007/BF01584650
[19]L.T. Watson, ”A variational approach to the linear complementarity problem”, Doctoral Dissertation, Department of Mathematics, University of Michigan (Ann Abor, MI, 1974).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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