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On a Smooth Dual Gap Function for a Class of Player Convex Generalized Nash Equilibrium Problems

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Listed:
  • Nadja Harms

    (University of Würzburg)

  • Tim Hoheisel

    (University of Würzburg)

  • Christian Kanzow

    (University of Würzburg)

Abstract

We consider a class of generalized Nash equilibrium problems, where both objective functions and constraints are allowed to depend on the decision variables of the other players. It is well known that this problem can be reformulated as a constrained optimization problem via the (regularized) Nikaido–Isoda-function, but this reformulation is usually nonsmooth. Here we observe that, under suitable conditions, this reformulation turns out to be the difference of two convex functions. This allows the application of the Toland-Singer duality theory in order to obtain a dual formulation, which provides an unconstrained and continuously differentiable optimization reformulation of the generalized Nash equilibrium problem. Moreover, based on a result from parametric optimization, the gradient of the unconstrained objective function is shown to be piecewise smooth. Some numerical results are presented to illustrate the theory.

Suggested Citation

  • Nadja Harms & Tim Hoheisel & Christian Kanzow, 2015. "On a Smooth Dual Gap Function for a Class of Player Convex Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 659-685, August.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:2:d:10.1007_s10957-014-0631-6
    DOI: 10.1007/s10957-014-0631-6
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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.
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    7. K. Kubota & M. Fukushima, 2010. "Gap Function Approach to the Generalized Nash Equilibrium Problem," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 511-531, March.
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