Abstract. We consider the solution of generalized Nash equilibrium problems by concatenating the KKT optimality conditions of each player's optimization problem into a single KKT-like system. We then propose two approaches for solving this KKT system. The first approach is rather simple and uses a merit-function/equation-based technique for the solution of the KKT system. The second approach, partially motivated by the shortcomings of the first one, is an interior-point-based method. We show that this second approach has strong theoretical properties and, in particular, that it is possible to establish global convergence under sensible conditions, this probably being the first result of its kind in the literature. We discuss the results of an extensive numerical testing on four KKT-based solution algorithms, showing that the new interior-point method is efficient and very robust.

We present a new algorithm for the solution of Generalized Nash Equilibrium Problems. This hybrid method combines the robustness of a potential reduction algorithm and the local quadratic convergence rate of the LP-Newton method. We base our local convergence theory on a local error bound and provide a new sufficient condition for it to hold that is weaker than known ones. In particular, this condition implies neither local uniqueness of a solution nor strict complementarity. We also report promising numerical results. © 2013 Springer Science+Business Media New York

SummaryWe consider optimal control problems with ordinary differential equations that are coupled by shared, possibly nonconvex, constraints. For these problems, we use the generalized Nash equilibrium approach and provide a reformulation of normalized Nash equilibria as solutions to a single optimal control problem. By this reformulation, we are able to prove existence, and in some settings, exploiting convexity properties, we also get a limited number or even uniqueness of the normalized Nash equilibria. Then, we use our approach to discuss traffic scenarios with several autonomous vehicles, whose dynamics is described through differential equations, and the avoidance of collisions couples the optimal control problems of the vehicles. For the solution to the discretized problems, we prove strong convergence of the states and weak convergence of the controls. Finally, using existing optimal control software, we show that the generalized Nash equilibrium approach leads to reasonable results for a crossing scenario with different vehicle models. KEYWORDS autonomous vehicles, existence of an equilibrium, generalized Nash equilibrium problem, optimal control, ordinary differential equations, traffic scenarios

Generalized Nash equilibrium problem, Jointly convex, Optimization reformulation, Continuity, PC 1 mapping, Semismoothness, Constant rank constraint qualification,

Using a regularized Nikaido-Isoda function, we present a (nonsmooth) constrained optimization reformulation of the player convex generalized Nash equilibrium problem (GNEP). Further we give an unconstrained reformulation of a large subclass of player convex GNEPs which, in particular, includes the jointly convex GNEPs. Both approaches characterize all solutions of a GNEP as minima of optimization problems. The smoothness properties of these optimization problems are discussed in detail, and it is shown that the corresponding objective functions are continuous and piecewise continuously differentiable under mild assumptions. Some numerical results based on the unconstrained optimization reformulation being applied to player convex GNEPs are also included.

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