×

On a subclass of \(P_ 0\). (English) Zbl 0831.15013

From authors’ summary: “We focus on a particular subclass of \(P_0\) matrices that has arisen within the field of linear complementarity theory, namely the class SU of sufficient matrices. Our principal result is that SU contains another class, \(P_*\). We conjecture that the latter two classes are in fact the same.”
Note: The above conjecture has been shown to be true by H. Väliaho [Linear Algebra Appl. 233, 109-129 (1995)].
Reviewer: A.Lal (Allahabad)

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
Full Text: DOI

References:

[1] Aganagić, M.; Cottle, R. W., A constructive characterization of \(Q_0\)-matrices with nonnegative principal minors, Math. Programming, 37, 223-231 (1987) · Zbl 0618.90091
[2] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic: Academic New York · Zbl 0484.15016
[3] Van Bokhoven, W. M.G.; Jess, J. A.G., Some new aspects of \(P_0\)- and \(P\)-matrices and their application to networks with ideal diodes, (Proceedings of the IEEE ISCAS (1978)), 806-810
[4] Van Bokhoven, W. M.G., Piecewise-Linear Modelling and Analysis (1981), Kluwer Technische Boeken: Kluwer Technische Boeken Deventer, Netherlands
[5] (Cottle, R. W.; Giannessi, F.; Lions, J.-L., Variational Inequalities and Complementary Problems: Theory and Applications (1980), Wiley: Wiley New York)
[6] Cottle, R. W.; Guu, S. M., Two characterizations of sufficient matrices, Linear Algebra Appl., 170, 65-74 (1992) · Zbl 0756.90087
[7] Cottle, R. W.; Pang, J. S.; Stone, R. E., The Linear Complementarity Problem (1992), Academic: Academic Boston · Zbl 0757.90078
[8] Cottle, R. W.; Pang, J. S.; Venkateswaran, V., Sufficient matrices and the linear complementarity problem, Linear Algebra Appl., 114/115, 231-249 (1989) · Zbl 0674.90092
[9] Cottle, R. W.; Stone, R. E., On the uniqueness of solutions to linear complementarity problems, Math. Programming, 27, 191-213 (1983) · Zbl 0516.90071
[10] Fiedler, M.; Pták, V., On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak Math. J., 12, 382-400 (1962) · Zbl 0131.24806
[11] Fiedler, M.; Pták, V., Some generalizations of positive definiteness and monotonicity, Numer. Math., 9, 163-172 (1966) · Zbl 0148.25801
[12] Gale, D.; Nikaido, H., The Jacobian matrix and global univalence of mappings, Math. Ann., 159, 81-93 (1965) · Zbl 0158.04903
[13] Harker, P. T.; Pang, J. S., Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming Ser. B, 48, 161-220 (1990) · Zbl 0734.90098
[14] Ingleton, A. W., A problem in linear inequalities, (Proc. London Math. Soc., 16 (1966)), 519-536 · Zbl 0166.03005
[15] Katzenelson, J., An algorithm for solving nonlinear resistor networks, Bell System Tech. J., 44, 1605-1620 (1965) · Zbl 0139.11404
[16] Kojima, M.; Megiddo, N.; Noma, T.; Yoshise, A., A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0745.90069
[17] Murty, K. G., Linear Complementarity, Linear and Nonlinear Programming (1988), Helderman: Helderman Berlin · Zbl 0634.90037
[18] Samelson, H.; Thrall, R. M.; Wesler, O., A partition theorem for Euclidean \(n\)-space, (Proc. Amer. Math. Soc., 9 (1958)), 805-807 · Zbl 0117.37901
[19] Sandberg, I. W.; Willson, A. N., Some theorems on properties of dc equations of nonlinear networks, Bell System Tech. J., 48, 1-34 (1969)
[20] Tucker, A. W., Principal pivotal transforms of square matrices, SIAM Rev., 5, 305 (1963)
[21] Willson, A. N., A useful generalization of the \(P_0\) matrix concept, Numer. Math., 17, 62-70 (1971) · Zbl 0199.49703
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.