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A differentiable path-following algorithm for computing perfect stationary points

Author

Listed:
  • Yang Zhan

    (City University of Hong Kong)

  • Peixuan Li

    (City University of Hong Kong)

  • Chuangyin Dang

    (City University of Hong Kong)

Abstract

This paper is concerned with the computation of perfect stationary point, which is a strict refinement of stationary point. A differentiable homotopy method is developed for finding perfect stationary points of continuous functions on convex polytopes. We constitute an artificial problem by introducing a continuously differentiable function of an extra variable. With the optimality conditions of this problem and a fixed point argument, a differentiable homotopy mapping is constructed. As the extra variable becomes close to zero, the homotopy path naturally provides a sequence of closely approximate stationary points on perturbed polytopes, and converges to a perfect stationary point on the original polytope. Numerical experiments are implemented to further illustrate the effectiveness of our method.

Suggested Citation

  • Yang Zhan & Peixuan Li & Chuangyin Dang, 2020. "A differentiable path-following algorithm for computing perfect stationary points," Computational Optimization and Applications, Springer, vol. 76(2), pages 571-588, June.
    • Handle: RePEc:spr:coopap:v:76:y:2020:i:2:d:10.1007_s10589-020-00181-3
    DOI: 10.1007/s10589-020-00181-3
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    1. Elzen, A. van den & Laan, G. van der & Talman, A.J.J., 1989. "An adjustment process for an exchange economy with linear production technologies," Serie Research Memoranda 0082, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    2. Yang Zhan & Chuangyin Dang, 2018. "A smooth path-following algorithm for market equilibrium under a class of piecewise-smooth concave utilities," Computational Optimization and Applications, Springer, vol. 71(2), pages 381-402, November.
    3. P. Herings & Karl Schmedders, 2006. "Computing equilibria in finance economies with incomplete markets and transaction costs," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 27(3), pages 493-512, April.
    4. Zhengyong Zhou & Bo Yu, 2014. "A smoothing homotopy method for variational inequality problems on polyhedral convex sets," Journal of Global Optimization, Springer, vol. 58(1), pages 151-168, January.
    5. Srihari Govindan & Tilman Klumpp, 2003. "Perfect equilibrium and lexicographic beliefs," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(2), pages 229-243.
    6. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 151-159, February.
    7. Qing Xu & Bo Yu & Guo-Chen Feng, 2005. "Homotopy Methods for Solving Variational Inequalities in Unbounded Sets," Journal of Global Optimization, Springer, vol. 31(1), pages 121-131, January.
    8. Antoon van den Elzen & Gerard van der Laan & Dolf Talman, 1994. "An Adjustment Process for an Economy with Linear Production Technologies," Mathematics of Operations Research, INFORMS, vol. 19(2), pages 341-351, May.
    9. P. Herings & Ronald Peeters, 2010. "Homotopy methods to compute equilibria in game theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 119-156, January.
    10. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(5), pages 687-698, October.
    11. 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.
    12. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(3), pages 381-386, June.
    13. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(4), pages 525-537, August.
    14. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(2), pages 285-292, April.
    15. P. Jean-Jacques Herings & Ronald J.A.P. Peeters, 2001. "symposium articles: A differentiable homotopy to compute Nash equilibria of n -person games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 18(1), pages 159-185.
    16. Eaves, B. Curtis & Schmedders, Karl, 1999. "General equilibrium models and homotopy methods," Journal of Economic Dynamics and Control, Elsevier, vol. 23(9-10), pages 1249-1279, September.
    17. Z. Lin & Y. Li, 1999. "Homotopy Method for Solving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 100(1), pages 207-218, January.
    18. Eaves, B. Curtis, 1976. "A finite algorithm for the linear exchange model," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 197-203, July.
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    Cited by:

    1. Zhan, Yang & Dang, Chuangyin, 2021. "Determination of general equilibrium with incomplete markets and default penalties," Journal of Mathematical Economics, Elsevier, vol. 92(C), pages 49-59.
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