Abstract
It is well known that the study of the Linear Complementarity Problem can be linked with the study of an appropriate quadratic programming problem. The solution set to a given LCP always contains the KKT-set of the corresponding quadratic programming problem. The reverse implication holds only for some very special classes of matrices as positive semidefinite, P-, row adequate, row sufficient and positive subdefinite (PSBD)matrices. The paper extends the concept of PSBD matrices and proves that, the above coincidence still holds for this larger class of matrices.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Y. Chabrillac and J.-P. Crouzeix, Definiteness and semidefiniteness of quad-ratic forms revisited, Linear Algebra and its Applications,63 (1984) 283–292.
W. Cottle, Manifestations of the Schur complement, Linear Algebra and its Applications,8 (1974) 189–211.
W. Cottle, J. S. Pang and V. Venkateswaran, Sufficient matrices and the lin-ear complementarity problem, Linear Algebra and its Applications, 114/115 (1989) 231–249.
W. Cottle, J. S. Pang and R. E. Stone, The linear complementarity problern, Academic Press, New York, 1992.
J.-P. Crouzeix, Contribution 6. l’étude des fonctions quasi-convexes, Thèse d’Etat, U.E.R des Sciences Exactes et Naturelles, Université de Clermont-Ferrand II, 1977.
J.-P. Crouzeix and J. A. Ferland, Criteria for quasiconvexity and pseudocon-vexity: relationships and comparisons, Mathernatical Programming,23 (1982) 193–205.
J.-P. Crouzeix and S. Schaible, Generalized monotone affine maps, SIAM J. Matrix Anal. Appl.,17 (1996) 992–997.
J.-P. Crouzeix, P. Marcotte and D. Zhu, Conditions ensuring the applicabil-ity of cutting-plane methods for solving variational inequalities, Mathernatical Programming,88 (2000) 521–539.
J.-P. Crouzeix, Characterizations of generalized convexity and generalized monotonicity, in: Generalized Convexity, Generalized Monotonicity: Recent Results (eds. J.-P. Crouzeix, J.-E. Martinez Legaz and M. Voile), Kluwer Academic Publishers, Dordrecht, 1998, pp. 237–256.
J.-P. Crouzeix, A. Hassouni, A. Lahlou and S. Schaible, Positive Sub-Definite Matrices, Generalized Monotonicity and Linear Complementarity Problems, To appear in SIAM J. Matrix Anal. Appl.
J.-P. Crouzeix, Manuscript, September 1998.
J.-A. Ferland, Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions, Linear Algebra and its Applications, 38 (1981) 51–63.
M. Fiedler and V. Pták, On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak Math. J., 12 (1962) 382–400.
M. Frank and P. Wolfe, An algorithm for quadratic programming, Naval Res. Logistic Quart, 3 (1956), 992–997.
A. W. Ingleton, The linear complementarity problem, J. London Math. Soc, 2 (1970) 330–336.
S. Komlósi, On pseudoconvex functions, Acta Sci. Math. (Szeged) 57 (1993) 569–586.
B. Martos, Sub-definite matrices and quadratic forms, SIAM J. Appl. Math., 17 (1969) 1215–1223.
H. Samelson, R. M. Thrall and O. Wesler, A partition theorem for Euclidean n-space, Proc. Amer. Math. Soc. 9 (1958) 805–807.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
Crouzeix, JP., Komlósi, S. (2001). The Linear Complementarity Problem and the Class of Generalized Positive Subdefinite Matrices. In: Giannessi, F., Pardalos, P., Rapcsák, T. (eds) Optimization Theory. Applied Optimization, vol 59. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0295-7_4
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0295-7_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4020-0009-6
Online ISBN: 978-1-4613-0295-7
eBook Packages: Springer Book Archive