[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/p/tin/wpaper/20090062.html
   My bibliography  Save this paper

Solving Discrete Systems of Nonlinear Equations

Author

Listed:
  • Gerard van der Laan

    (VU University, Amsterdam, The Netherlands)

  • Dolf Talman

    (Tilburg University, The Netherlands)

  • Zaifu Yang

    (Yokohama National University, Japan)

Abstract
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.

Suggested Citation

  • Gerard van der Laan & Dolf Talman & Zaifu Yang, 2009. "Solving Discrete Systems of Nonlinear Equations," Tinbergen Institute Discussion Papers 09-062/1, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20090062
    as

    Download full text from publisher

    File URL: https://papers.tinbergen.nl/09062.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. P. J. J. Herings & G. A. Koshevoy & A. J. J. Talman & Z. Yang, 2004. "General Existence Theorem of Zero Points," Journal of Optimization Theory and Applications, Springer, vol. 120(2), pages 375-394, February.
    3. 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.
    4. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    5. Zaifu Yang, 2008. "On the Solutions of Discrete Nonlinear Complementarity and Related Problems," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 976-990, November.
    6. 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.
    7. 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.
    8. 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.
    9. G. van der Laan, 1981. "Simplicial fixed point algorithms," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 35(1), pages 58-58, March.
    10. R. Saigal, 1983. "A Homotopy for Solving Large, Sparse and Structured Fixed Point Problems," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 557-578, November.
    11. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
    12. Iimura, Takuya, 2003. "A discrete fixed point theorem and its applications," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 725-742, September.
    13. Iimura, Takuya & Murota, Kazuo & Tamura, Akihisa, 2005. "Discrete fixed point theorem reconsidered," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 1030-1036, December.
    14. 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.
    15. 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.
    16. Peter M. Reiser, 1981. "A Modified Integer Labeling for Complementarity Algorithms," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 129-139, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kazuo Murota, 2016. "Discrete convex analysis: A tool for economics and game theory," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 1(1), pages 151-273, December.
    2. Michael J. Todd, 2016. "Computation, Multiplicity, and Comparative Statics of Cournot Equilibria in Integers," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 1125-1134, August.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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.
    2. G. Laan & A. J. J. Talman & Z. Yang, 2010. "Combinatorial Integer Labeling Theorems on Finite Sets with Applications," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 391-407, February.
    3. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2007. "Combinatorial Integer Labeling Thorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations," Other publications TiSEM 264c28a5-10b6-44e1-9694-4, Tilburg University, School of Economics and Management.
    4. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2005. "Solving Discrete Zero Point Problems with Vector Labeling," Other publications TiSEM 9bd940ee-3fe6-4201-aede-7, Tilburg University, School of Economics and Management.
    5. 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.
    6. Dolf Talman & Zaifu Yang, 2012. "On a Parameterized System of Nonlinear Equations with Economic Applications," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 644-671, August.
    7. 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.
    8. P. J. J. Herings & G. A. Koshevoy & A. J. J. Talman & Z. Yang, 2004. "General Existence Theorem of Zero Points," Journal of Optimization Theory and Applications, Springer, vol. 120(2), pages 375-394, February.
    9. Kazuo Murota, 2016. "Discrete convex analysis: A tool for economics and game theory," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 1(1), pages 151-273, December.
    10. Talman, A.J.J. & Yang, Z.F., 2004. "The Computation of a Coincidence of Two Mappings," Other publications TiSEM 9fbcc219-da4d-4564-b4e3-6, Tilburg University, School of Economics and Management.
    11. Hoang Ngoc Tuan, 2015. "Boundedness of a Type of Iterative Sequences in Two-Dimensional Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 234-245, January.
    12. Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004. "Solving Discrete Zero Point Problems," Tinbergen Institute Discussion Papers 04-112/1, Tinbergen Institute.
    13. Talman, A.J.J. & Yang, Z.F., 2003. "On the Connectedness of Coincidences and Zero Points of Mappings," Discussion Paper 2003-73, Tilburg University, Center for Economic Research.
    14. Doup, T.M. & van der Laan, G. & Talman, A.J.J., 1984. "The (2n+1-2)-ray algorithm : A new simplicial algorithm to compute economic equilibria," Research Memorandum FEW 151, Tilburg University, School of Economics and Management.
    15. Chuangyin Dang & Hans van Maaren, 1998. "A Simplicial Approach to the Determination of an Integer Point of a Simplex," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 403-415, May.
    16. Ruys, P.H.M. & van der Laan, G., 1987. "Computation of an industrial equilibrium," Research Memorandum FEW 257, Tilburg University, School of Economics and Management.
    17. C. Dang, 2001. "Computing an Integer Point of a Class of Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 108(2), pages 333-348, February.
    18. Zaifu Yang, 2008. "On the Solutions of Discrete Nonlinear Complementarity and Related Problems," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 976-990, November.
    19. Gue Myung Lee & Tiến-Sơn Phạm, 2016. "Stability and Genericity for Semi-algebraic Compact Programs," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 473-495, May.
    20. G. M. Lee & N. N. Tam & N. D. Yen, 2006. "Continuity of the Solution Map in Quadratic Programs under Linear Perturbations," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 415-423, June.

    More about this item

    Keywords

    Discrete system of equations; triangulation; simplicial algorithm; fixed point; zero point;
    All these keywords.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:tin:wpaper:20090062. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Tinbergen Office +31 (0)10-4088900 (email available below). General contact details of provider: https://edirc.repec.org/data/tinbenl.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.