Theory and Solution Methods for Generalized Nash Equilibrium Problems Governed by Networks of Nonlinear Hyperbolic Conservation Laws
The aim of this project is the analysis of (Generalized) Nash Equilibrium Problems ((G)NEPs) that are governed by networks of nonlinear hyperbolic conservation or balance laws as well as the development and analysis of efficient solution methods for these problems. Networks of conservation laws are an active research field and have led to innovative models of flow or transport problems, e.g., for traffic networks, supply chains, data networks and water or gas networks. In all of these applications, (G)NEPs provide powerful models for the interaction of multiple non-cooperative agents who optimize their strategies. Since solutions of conservation laws may develop discontinuities, they exhibit additional nonsmooth phenomena which in combination with games fits perfectly to the research topics of SPP 1962. Based on recent results concerning the existence and stability of solutions of networks of conservation laws as well as the optimal control of conservation laws we will develop an analytical setting that yields stability and differentiability properties of the players' cost functionals. Moreover, we will derive an adjoint-based derivative representation. This will be used to study the existence of quasi-Nash equilibria (QNE) for nonconvex NEPs as well as QNE and variational equilibria (VE) for nonconvex GNEPs of this type. Here QNE and VE are characterized by variational inequalities that aggregate the players' first order optimality systems. For games with convex feasible sets, the relation of QNE and VE to global minima of merit functions based on regularized Nikaido-Isoda functions will be investigated and used to study proximal best response maps for proving existence results. Since the considered games are nonconvex, we plan to establish differentiability of these merit functions and to develop globally convergent descent methods for convexly constrained (G)NEPs. In the case of nonconvex constraints, in particular state constraints, QNE / VE concepts based on Lagrange multipliers and suitable constraint qualifications will be derived and the existence of equilibria will be studied. For (G)NEPs with nonconvex constraints we will investigate augmented Lagrangian methods that approximately solve a sequence of convexly constrained (G)NEPs, to which the above class of globally convergent methods can be applied. Ways for accelerating these descent methods by nonsmooth Newton steps as well as ideas for a decomposition via block iterations will be explored. Although the methods are inspired by an analytical setting for games governed by hyperbolic networks they will be designed to cover other PDE-constrained games as well. The developed methods will be implemented and tested for games in traffic flow and supply chain models.
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