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Solving Discrete Zero Point Problems with Vector Labeling

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Author Info

  • Gerard van der Laan

    ()
    (Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam)

  • Dolf Talman

    ()
    (Department of Econometrics & Operations Research, and CentER, Tilburg University)

  • Zaifu Yang

    ()
    (Faculty of Business Administration, Yokohama University)

Abstract

In this paper we present two general results on the existence of a discrete zero point of a function from the n-dimensional integer lattice Zn to the n-dimensional Euclidean space Rn. Under two different boundary conditions, we give a constructive proof using a combinatorial argument based on a simplicial algorithm with vector labeling and lexicographic linear programming pivot steps. We also adept the algorithm to prove the existence of a solution to the discrete complementarity problem.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 05-106/1.

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Date of creation: 28 Nov 2005
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Handle: RePEc:dgr:uvatin:20050106

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Web page: http://www.tinbergen.nl

Related research

Keywords: integer lattice; zero point; vector labeling rule; simplicial algorithm; Borsuk-Ulam; discrete complementarity;

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References

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  1. Gerard van der Laan & Dolf Talman & Zaifu Yang, 2005. "Computing Integral Solutions of Complementarity Problems," Tinbergen Institute Discussion Papers 05-006/1, Tinbergen Institute.
  2. 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.
  3. Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004. "Solving Discrete Zero Point Problems," Tinbergen Institute Discussion Papers 04-112/1, Tinbergen Institute.
  4. C. E. Lemke, 1965. "Bimatrix Equilibrium Points and Mathematical Programming," Management Science, INFORMS, vol. 11(7), pages 681-689, May.
  5. 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.
  6. Talman, A.J.J. & Laan , G. van der, 1987. "Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153048, Tilburg University.
  7. Iimura, Takuya, 2003. "A discrete fixed point theorem and its applications," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 725-742, September.
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Cited by:
  1. Talman, Dolf & Yang, Zaifu, 2009. "A discrete multivariate mean value theorem with applications," European Journal of Operational Research, Elsevier, vol. 192(2), pages 374-381, January.

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