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The Cone Condition and Nonsmoothness in Linear Generalized Nash Games

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  • Oliver Stein

    (Karlsruhe Institute of Technology)

  • Nathan Sudermann-Merx

    (Karlsruhe Institute of Technology)

Abstract

We consider linear generalized Nash games and introduce the so-called cone condition, which characterizes the smoothness of a gap function that arises from a reformulation of the generalized Nash equilibrium problem as a piecewise linear optimization problem based on the Nikaido–Isoda function. Other regularity conditions such as the linear independence constraint qualification or the strict Mangasarian–Fromovitz condition are only sufficient for smoothness, but have the advantage that they can be verified more easily than the cone condition. Therefore, we present special cases, where these conditions are not only sufficient, but also necessary for smoothness of the gap function. Our main tool in the analysis is a global extension of the gap function that allows us to overcome the common difficulty that its domain may not cover the whole space.

Suggested Citation

  • Oliver Stein & Nathan Sudermann-Merx, 2016. "The Cone Condition and Nonsmoothness in Linear Generalized Nash Games," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 687-709, August.
    • Handle: RePEc:spr:joptap:v:170:y:2016:i:2:d:10.1007_s10957-015-0779-8
    DOI: 10.1007/s10957-015-0779-8
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    References listed on IDEAS

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    1. Axel Dreves & Christian Kanzow & Oliver Stein, 2012. "Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems," Journal of Global Optimization, Springer, vol. 53(4), pages 587-614, August.
    2. Axel Dreves, 2014. "Finding all solutions of affine generalized Nash equilibrium problems with one-dimensional strategy sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(2), pages 139-159, October.
    3. A. Izmailov & M. Solodov, 2015. "Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 1-26, April.
    4. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    5. Anna Heusinger & Christian Kanzow, 2009. "Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions," Computational Optimization and Applications, Springer, vol. 43(3), pages 353-377, July.
    6. A. Izmailov & M. Solodov, 2015. "Rejoinder on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 48-52, April.
    7. Oliver Stein & Nathan Sudermann-Merx, 2014. "On smoothness properties of optimal value functions at the boundary of their domain under complete convexity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 79(3), pages 327-352, June.
    8. Daniel Ralph & Oliver Stein, 2011. "The C-Index: A New Stability Concept for Quadratic Programs with Complementarity Constraints," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 504-526, August.
    9. William W. Cooper & Lawrence M. Seiford & Kaoru Tone, 2007. "Data Envelopment Analysis," Springer Books, Springer, edition 0, number 978-0-387-45283-8, September.
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    Cited by:

    1. Stein, Oliver & Sudermann-Merx, Nathan, 2018. "The noncooperative transportation problem and linear generalized Nash games," European Journal of Operational Research, Elsevier, vol. 266(2), pages 543-553.
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    3. Simone Sagratella, 2017. "Algorithms for generalized potential games with mixed-integer variables," Computational Optimization and Applications, Springer, vol. 68(3), pages 689-717, December.

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