IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v76y2020i2d10.1007_s10589-020-00181-3.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-020-00181-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10589-020-00181-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 151-159, February.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(3), pages 381-386, June.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(5), pages 687-698, October.
    14. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(4), pages 525-537, August.
    15. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(2), pages 285-292, April.
    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.
    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. 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.
    2. Chuangyin Dang & P. Jean-Jacques Herings & Peixuan Li, 2022. "An Interior-Point Differentiable Path-Following Method to Compute Stationary Equilibria in Stochastic Games," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1403-1418, May.

    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. Dang, Chuangyin & Meng, Xiaoxuan & Talman, Dolf, 2015. "An Interior-Point Path-Following Method for Computing a Perfect Stationary Point of a Polynomial Mapping on a Polytope," Other publications TiSEM 07b7a0e7-f814-4ec2-a3a7-e, Tilburg University, School of Economics and Management.
    2. Yang Zhan & Chuangyin Dang, 2021. "Computing equilibria for markets with constant returns production technologies," Annals of Operations Research, Springer, vol. 301(1), pages 269-284, June.
    3. Chuangyin Dang & P. Jean-Jacques Herings & Peixuan Li, 2022. "An Interior-Point Differentiable Path-Following Method to Compute Stationary Equilibria in Stochastic Games," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1403-1418, May.
    4. Talman, A.J.J. & Yamamoto, M., 2001. "Contiuum of Zero Points of a Mapping on a Compact Convex Set," Other publications TiSEM 57411440-5b14-448e-8c27-3, Tilburg University, School of Economics and Management.
    5. 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.
    6. Dang, Chuangyin & Herings, P. Jean-Jacques & Li, Peixuan, 2020. "An Interior-Point Path-Following Method to Compute Stationary Equilibria in Stochastic Games," Research Memorandum 001, Maastricht University, Graduate School of Business and Economics (GSBE).
    7. Cao, Yiyin & Dang, Chuangyin & Xiao, Zhongdong, 2022. "A differentiable path-following method to compute subgame perfect equilibria in stationary strategies in robust stochastic games and its applications," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1032-1050.
    8. 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.
    9. Allen C. Goodman & Miron Stano, 2000. "Hmos and Health Externalities: A Local Public Good Perspective," Public Finance Review, , vol. 28(3), pages 247-269, May.
    10. Bettina Campedelli & Andrea Guerrina & Giulia Romano & Chiara Leardini, 2014. "La performance della rete ospedaliera pubblica della regione Veneto. L?impatto delle variabili ambientali e operative sull?efficienza," MECOSAN, FrancoAngeli Editore, vol. 2014(92), pages 119-142.
    11. Penn Loh & Zoë Ackerman & Joceline Fidalgo & Rebecca Tumposky, 2022. "Co-Education/Co-Research Partnership: A Critical Approach to Co-Learning between Dudley Street Neighborhood Initiative and Tufts University," Social Sciences, MDPI, vol. 11(2), pages 1-17, February.
    12. O'Brien, Raymond & Patacchini, Eleonora, 2003. "Testing the exogeneity assumption in panel data models with "non classical" disturbances," Discussion Paper Series In Economics And Econometrics 0302, Economics Division, School of Social Sciences, University of Southampton.
    13. YongSeog Kim & W. Nick Street & Gary J. Russell & Filippo Menczer, 2005. "Customer Targeting: A Neural Network Approach Guided by Genetic Algorithms," Management Science, INFORMS, vol. 51(2), pages 264-276, February.
    14. Yanling Li & Zita Oravecz & Shuai Zhou & Yosef Bodovski & Ian J. Barnett & Guangqing Chi & Yuan Zhou & Naomi P. Friedman & Scott I. Vrieze & Sy-Miin Chow, 2022. "Bayesian Forecasting with a Regime-Switching Zero-Inflated Multilevel Poisson Regression Model: An Application to Adolescent Alcohol Use with Spatial Covariates," Psychometrika, Springer;The Psychometric Society, vol. 87(2), pages 376-402, June.
    15. Oscar J. Cacho & Robyn L. Hean & Russell M. Wise, 2003. "Carbon‐accounting methods and reforestation incentives," Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 47(2), pages 153-179, June.
    16. Walter M. Cadette, 1999. "Financing Long-Term Care: Options for Policy," Economics Working Paper Archive wp_283, Levy Economics Institute.
    17. Eggli, Yves & Halfon, Patricia & Chikhi, Mehdi & Bandi, Till, 2006. "Ambulatory healthcare information system: A conceptual framework," Health Policy, Elsevier, vol. 78(1), pages 26-38, August.
    18. M. A. Noor & E.A. Al-Said, 2002. "Finite-Difference Method for a System of Third-Order Boundary-Value Problems," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 627-637, March.
    19. Yong He & Zhiyi Tan, 2002. "Ordinal On-Line Scheduling for Maximizing the Minimum Machine Completion Time," Journal of Combinatorial Optimization, Springer, vol. 6(2), pages 199-206, June.
    20. Henderson, James E. & Dunn, Michael A., 2007. "Investigating the Potential of Fee-Based Recreation on Private Lands in the Lower Mississippi River Delta," 2007 Annual Meeting, February 4-7, 2007, Mobile, Alabama 34822, Southern Agricultural Economics Association.

    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:spr:coopap:v:76:y:2020:i:2:d:10.1007_s10589-020-00181-3. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.