Equilibrium Problems with Complementarity Constraints in Banach Spaces:Stationarity Concepts, Applications and Algorithms

Abstract

In mathematics, the field of non-cooperative game theory models the competition between several parties, which are called players. Therein, each player tries to reach an individual goal, which is described by an optimization problem. However and in contrast to classical nonlinear programming, there exists a dependency between the players, i.e. the choice of a suitable strategy influences the behavior and the reward of the player’s opponents and vice versa. For this reason, a popular solution concept is given by Nash equilibria, which were introduced by John Forbes Nash in his Ph.D. thesis in 1950. In order to prove the existence of a Nash equilibrium, the convexity of the underlying optimization problem is a central requirement. However, this assumption does not hold in general. This thesis is devoted to special equilibrium problems in Banach spaces, which can be described by equilibrium problems with equilibrium/complementarity constraints (EPEC/EPCC). Due to the structure of the underlying feasible set, those games are nonconvex. Motivated by known results with respect to mathematical programs with complementarity constraints, we focus on weaker Nash equilibrium concepts, which can at least be seen as necessary conditions for a Nash equilibrium under suitable assumptions. In the first part of this work, we concentrate on multi-leader multi-follower games, where the participating players are divided hierarchically into leaders and followers, which compete on their particular level with each other. Under suitable assumptions, the solution of the lower level is described by its necessary and sufficient first-order optimality system and can be written as an EPCC. In this context, we first analyze the latter problem in abstract Banach spaces and afterwards, consider the special case of a multi-leader single-follower game (MLFG), where the lower level is given by a quadratic problem in a Hilbert space. For the latter one, we show on the basis of two known penalization techniques that there exist sequences of auxiliary equilibrium problems, which approximate the corresponding EPCC. In the following application that extends known contributions on an optimal control framework of the obstacle problem, we use these auxiliary games and show that both generate sequences, which converge at least to an ϵ-almost C-stationary Nash equilibrium of the original MLFG. The results are analyzed numerically on the basis of a Gaus-Seideltype algorithm and are tested with respect to two examples. The second part is motivated by the work [14] ”A generalized Nash equilibrium approach for optimal control problems of autonomous cars” by Axel Dreves and Matthias Gerdts, where a traffic scenario between several intelligent cars is modeled by a dynamic equilibrium problem. Due to the collision avoidance constraint, this game is non-convex. However, we show that it can be written as a generalized Nash equilibrium problem with mixed-integer variables (MINEP), which again is equivalent to an EPCC. In contrast to the first application, we now concentrate on problems in Lebesgue spaces. In the following, we compare known results from abstract Banach spaces and the corresponding ones in Lebesgue spaces. In particular, we show that for general MINEPs all weak Nash equilibrium concepts coincide. Based on these observations, we apply the results to the traffic scenario. In this context, we again use a penalization technique and deduce by the generated sequence of Nash equilibrium problems that we find a sequence of equilibrium points, which converge to an S-stationary Nash equilibrium of MINEP. We end up with a numerical analysis and test the results with two hypothetical traffic scenarios.