The Cone Condition and Nonsmoothness in Linear Generalized Nash Games
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.
KeywordsGeneralized Nash equilibrium problem Nikaido–Isoda function Piecewise linear function Constraint qualification Genericity Parametric optimization
Mathematics Subject Classification91A06 91A10 90C31
We thank the anonymous referees for their precise and substantial remarks, which helped to significantly improve the paper. Furthermore, we would like to thank Christian Kanzow and Axel Dreves for fruitful discussions on the subject of this paper. This research was partially supported by the DFG (Deutsche Forschungsgemeinschaft) under grant STE 772/13-1.
- 22.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., online first, (2014) doi: 10.1007/s10957-014-0631-6
- 27.Van Hang, N.T., Yen, N.D.: On the problem of minimizing a difference of polyhedral convex functions under linear constraints. J. Optim. Theory Appl., online first, (2015). doi: 10.1007/s10957-015-0769-x