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Solving discrete systems of nonlinear equations

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  • van der Laan, Gerard
  • Talman, Dolf
  • Yang, Zaifu

Abstract

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 and each simplex of the triangulation lies in an n-dimensional 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.

Suggested Citation

  • van der Laan, Gerard & Talman, Dolf & Yang, Zaifu, 2011. "Solving discrete systems of nonlinear equations," European Journal of Operational Research, Elsevier, vol. 214(3), pages 493-500, November.
  • Handle: RePEc:eee:ejores:v:214:y:2011:i:3:p:493-500
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    References listed on IDEAS

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    1. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2005. "Computing Integral Solutions of Complementarity Problems," Discussion Paper 2005-5, Tilburg University, Center for Economic Research.
    2. Talman, A.J.J. & Yang, Z.F., 2009. "A discrete multivariate mean value theorem with applications," Other publications TiSEM d48f2a19-dcc2-40e4-9085-5, Tilburg University, School of Economics and Management.
    3. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2007. "A vector labeling method for solving discrete zero point and complementarity problems," Other publications TiSEM 070869d0-4e42-4d34-85f9-b, Tilburg University, School of Economics and Management.
    4. Herings P. Jean-Jacques & Koshevoy Gleb A. & Talman Dolf & Yang Zaifu, 2002. "A General Existence Theorem of Zero Points," Research Memorandum 055, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    5. 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.
    6. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Discussion Paper 1999-72, Tilburg University, Center for Economic Research.
    7. Iimura, Takuya, 2003. "A discrete fixed point theorem and its applications," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 725-742, September.
    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.
    10. Talman, A.J.J. & van der Laan, G., 1979. "A restart algorithm for computing fixed points without an extra dimension," Other publications TiSEM 1f2102f8-e6da-4e9c-a2ed-9, Tilburg University, School of Economics and Management.
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    More about this item

    Keywords

    Discrete system of equations Triangulation Simplicial algorithm Fixed point Zero point;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C68 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computable General Equilibrium Models
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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