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

Nadja Harms (), Tim Hoheisel () and Christian Kanzow ()
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Nadja Harms: University of Würzburg
Tim Hoheisel: University of Würzburg
Christian Kanzow: University of Würzburg

Journal of Optimization Theory and Applications, 2015, vol. 166, issue 2, No 17, 659-685

Abstract: 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.

Keywords: Generalized Nash equilibrium; DC optimization; Conjugate function; Dual gap function; Nonconvex duality; Optimal solution mapping; PC $$^1$$ 1 function; 49M29; 65K10; 90C33 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10957-014-0631-6

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