Existence and uniqueness of solutions for the generalized linear complementarity problem
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Abstract
Cottle and Dantzig (Ref. 1) showed that the generalized linear complementarity problem has a solution for anyq∈R m ifM is a vertical blockP-matrix of type (m1,...,m n ). They also extended known characterizations of squareP-matrices to vertical blockP-matrices.
Here we show, using a technique similar to Murty's (Ref. 2), that there exists a unique solution for anyq∈R m if and only ifM is a vertical blockP-matrix of type (m1,...,m n ). To obtain this characterization, we employ a generalization of Tucker's theorem (Ref. 3) and a generalization of a theorem initially introduced by Gale and Nikaido (Ref. 4).
Key Words
Linear complementarity problems generalized linear complementarity P-matrices Tucker's theoremPreview
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References
- 1.Cottle, R. W., andDantzig, G. B.,A Generalization of the Linear Complementarity Problem, Journal of Combinatorial Theory, Vol. 8, pp. 79–90, 1970.Google Scholar
- 2.Murty, K. G.,On a Characterization of P-Matrices, SIAM Journal on Applied Mathematics, Vol. 20, pp. 378–383, 1971.Google Scholar
- 3.Tucker, A. W.,Principal Pivot Transforms of Square Matrices, SIAM Review, Vol. 5, pp. 305, 1963.Google Scholar
- 4.Gale, D., andNikaido, H.,The Jacobian Matrix and Global Univalence of Mappings, Mathematische Analen, Vol. 159, pp. 81–93, 1965.Google Scholar
- 5.Cottle, R. W., andDantzig, G. B.,Complementarity Pivot Theory of Mathematical Programming, Linear Algebra and Its Applications, Vol. 1, pp. 103–125, 1968.Google Scholar
- 6.Lemke, C. E.,Recent Results on Complementarity Problems, Nonlinear Programming, Edited by J. B. Rosen, O. L. Mangasarian, and K. Ritter, Academic Press, New York, New York, 1970.Google Scholar
- 7.Oh, K. P.,The Formulation of the Mixed Lubrication Problem as a Generalized Nonlinear Complementarity Problem, Transactions of the ASME, Journal of Tribology, Vol. 108, pp. 598–604, 1986.Google Scholar
- 8.Murty, K. G.,On the Number of Solutions to the Complementarity Problem and Spanning Properties of Complementary Cones, Linear Algebra and Applications, Vol. 5, pp. 65–108, 1972.Google Scholar
- 9.Lemke, C. E.,On Complementary Pivot Theory, Mathematics of Decisions Sciences, Edited by G. B. Dantzig and A. F. Veinott, Jr., American Mathematical Society, Providence, Rhode Island, Vol. 1, pp. xxx-xxx, 1968.Google Scholar
- 10.Fiedler, M., andPtak, V.,On Matrices with Nonpositive Off-Diagonal Elements and Positive Principal Minors, Czechoslovakian Mathematics Journal, Vol. 12, pp. 382–400, 1962.Google Scholar
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© Plenum Publishing Corporation 1995