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

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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.

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Acknowledgments

This research was partially supported by the DFG (Deutsche Forschungsgemeinschaft) under Grant KA 1296/18-1.

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  1. Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, 97074, Würzburg, Germany

    Nadja Harms, Tim Hoheisel & Christian Kanzow

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  1. Nadja Harms
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  2. Tim Hoheisel
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  3. Christian Kanzow
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Correspondence to Christian Kanzow.

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Harms, N., Hoheisel, T. & Kanzow, C. On a Smooth Dual Gap Function for a Class of Player Convex Generalized Nash Equilibrium Problems. J Optim Theory Appl 166, 659–685 (2015). https://doi.org/10.1007/s10957-014-0631-6

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  • DOI: https://doi.org/10.1007/s10957-014-0631-6

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