Abstract. We consider the generalized Nash equilibrium problem which, in contrast to the standard Nash equilibrium problem, allows joint constraints of all players involved in the game. Using a regularized Nikaido-Isoda-function, we then present three optimization problems related to the generalized Nash equilibrium problem. The first optimization problem is a complete reformulation of the generalized Nash game in the sense that the global minima are precisely the solutions of the game. However, this reformulation is nonsmooth. We then modify this approach and obtain a smooth constrained optimization problem whose global minima correspond to so-called normalized Nash equilibria. The third approach uses the difference of two regularized Nikaido-Isoda-functions in order to get a smooth unconstrained optimization problem whose global minima are, once again, precisely the normalized Nash equilibria. Conditions for stationary points to be global minima of the two smooth optimization problems are also given. Some numerical results illustrate the behaviour of our approaches.
We consider the generalized Nash equilibrium problem (GNEP), where not only the players' cost functions but also their strategy spaces depend on the rivals' decision variables. Existence results for GNEPs are typically shown by using a fixed point argument for a certain set-valued function. Here we use a regularization of this set-valued function in order to obtain a single-valued function that is easier to deal with from a numerical point of view. We show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria. This fixed point formulation is then used to develop a nonsmooth Newton method for computing a normalized equilibrium. The method uses a so-called computable generalized Jacobian that is much easier to compute than Clarke generalized Jacobian or B-subdifferential. We establish local superlinear/quadratic convergence of the method under the constant rank constraint qualification, which is weaker than the frequently used linear independence constraint qualification, and a suitable 123 100 A. von Heusinger et al.second-order condition. Some numerical results are presented to illustrate the performance of the method.
Abstract. The generalized Nash equilibrium problem is a Nash game which, in contrast to the standard Nash equilibrium problem, allows the strategy sets of each player to depend on the decision variables of all other players. It was recently shown by the authors that this generalized Nash equilibrium problem can be reformulated as both an unconstrained and a constrained optimization problem with continuously differentiable objective functions. This paper further investigates these approaches and shows, in particular, that the objective functions are SC 1 -functions. Moreover, conditions for the local superlinear convergence of a semismooth Newton method being applied to the unconstrained optimization reformulation are also given. Some numerical results indicate that this method works quite well on a number of problems coming from different application areas.
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