The Cone Condition and Nonsmoothness in Linear Generalized Nash Games

Oliver Stein () and Nathan Sudermann-Merx ()
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Oliver Stein: Karlsruhe Institute of Technology
Nathan Sudermann-Merx: Karlsruhe Institute of Technology

Journal of Optimization Theory and Applications, 2016, vol. 170, issue 2, No 19, 687-709

Abstract: 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.

Keywords: Generalized Nash equilibrium problem; Nikaido–Isoda function; Piecewise linear function; Constraint qualification; Genericity; Parametric optimization; 91A06; 91A10; 90C31 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-015-0779-8

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