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Solving discrete systems of nonlinear equations

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  • Laan, G. van der
  • Talman, A.J.J.

    (Tilburg University)

  • Yang, Z.F.

    (Tilburg University)

Abstract

In this paper we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Zn of the n-dimensional Euclidean space IRn. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Zn and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangu- lation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the `continuity property' is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further adapt the main result to a discrete variant of the well-known Borsuk-Ulam theorem and to a theorem for the existence of a solution for the discrete nonlinear complementarity problem.

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Bibliographic Info

Paper provided by Tilburg University in its series Open Access publications from Tilburg University with number urn:nbn:nl:ui:12-4839550.

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Date of creation: 2011
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Publication status: Published in European Journal of Operational Research (2011) v.214, p.493-500
Handle: RePEc:ner:tilbur:urn:nbn:nl:ui:12-4839550

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Web page: http://www.tilburguniversity.edu/

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References

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  1. Talman, A.J.J. & Yang, Z.F., 2006. "A Discrete Multivariate Mean Value Theorem with Applications," Discussion Paper 2006-106, Tilburg University, Center for Economic Research.
  2. Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2007. "A vector labeling method for solving discrete zero point and complementarity problems," Open Access publications from Tilburg University urn:nbn:nl:ui:12-284192, Tilburg University.
  3. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 2001. "Variational inequality problems with a continuum of solutions: Existence and computation," Open Access publications from Tilburg University urn:nbn:nl:ui:12-86829, Tilburg University.
  4. Talman, A.J.J. & Laan, G. van der, 1979. "A restart algorithm for computing fixed points without an extra dimension," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153012, Tilburg University.
  5. Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2005. "Computing Integral Solutions of Complementarity Problems," Discussion Paper 2005-5, Tilburg University, Center for Economic Research.
  6. Iimura, Takuya, 2003. "A discrete fixed point theorem and its applications," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 725-742, September.
  7. Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004. "Solving Discrete Zero Point Problems," Tinbergen Institute Discussion Papers 04-112/1, Tinbergen Institute.
  8. Iimura, Takuya & Murota, Kazuo & Tamura, Akihisa, 2005. "Discrete fixed point theorem reconsidered," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 1030-1036, December.
  9. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
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