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**Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems.**
*(English)*
Zbl 1227.90040

Summary: Generalized Nash equilibrium problems (GNEPs) allow, in contrast to standard Nash equilibrium problems, a dependence of the strategy space of one player from the decisions of the other players. In this paper, we consider jointly convex GNEPs which form an important subclass of the general GNEPs. Based on a regularized Nikaido-Isoda function, we present two (nonsmooth) reformulations of this class of GNEPs, one reformulation being a constrained optimization problem and the other one being an unconstrained optimization problem. While most approaches in the literature compute only a so-called normalized Nash equilibrium, which is a subset of all solutions, our two approaches have the property that their minima characterize the set of all solutions of a GNEP. We also investigate the smoothness properties of our two optimization problems and show that both problems are continuous under a Slater-type condition and, in fact, piecewise continuously differentiable under the constant rank constraint qualification. Finally, we present some numerical results based on our unconstrained optimization reformulation.

### MSC:

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

91A10 | Noncooperative games |

90C25 | Convex programming |

### Keywords:

generalized Nash equilibrium problem; jointly convex; optimization reformulation; continuity; \(PC^{1}\) mapping; semismoothness; constant rank constraint qualification### Software:

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\textit{A. Dreves} and \textit{C. Kanzow}, Comput. Optim. Appl. 50, No. 1, 23--48 (2011; Zbl 1227.90040)

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### References:

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