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Solving Discrete Systems of Nonlinear Equations


Author Info

  • Gerard van der Laan

    (VU University, Amsterdam, The Netherlands)

  • Dolf Talman

    (Tilburg University, The Netherlands)

  • Zaifu Yang

    (Yokohama National University, Japan)


This discussion paper led to a publication in 'European Journal of Operational Research' , 214(3), 493-500. We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice of the n-dimensional Euclidean space. 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 the integer lattice and each simplex of the triangulation lies in a cube of size one. With respect to this triangulation 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 obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 09-062/1.

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Date of creation: 16 Jul 2009
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Handle: RePEc:dgr:uvatin:20090062

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Keywords: Discrete system of equations; triangulation; simplicial algorithm; fixed point; zero point;

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  1. 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.
  2. Iimura, Takuya, 2003. "A discrete fixed point theorem and its applications," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 725-742, September.
  3. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 216R, Cowles Foundation for Research in Economics, Yale University.
  4. Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2005. "Computing Integral Solutions of Complementarity Problems," Discussion Paper, Tilburg University, Center for Economic Research 2005-5, Tilburg University, Center for Economic Research.
  5. Talman, Dolf & Yang, Zaifu, 2009. "A discrete multivariate mean value theorem with applications," European Journal of Operational Research, Elsevier, Elsevier, vol. 192(2), pages 374-381, January.
  6. 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.
  7. 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.
  8. Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004. "Solving Discrete Zero Point Problems," Tinbergen Institute Discussion Papers, Tinbergen Institute 04-112/1, Tinbergen Institute.
  9. Iimura, Takuya & Murota, Kazuo & Tamura, Akihisa, 2005. "Discrete fixed point theorem reconsidered," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 1030-1036, December.
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