Theory and Solution Methods for Generalized Nash Equilibrium Problems Governed by Networks of Nonlinear Hyperbolic Conservation Laws

这个项目的目的是分析由非线性双曲型守恒律或平衡律网络控制的(广义)Nash均衡问题((G)Net)，并开发和分析这些问题的有效求解方法。保护法网络是一个活跃的研究领域，并导致了流量或运输问题的创新模型，例如交通网络、供应链、数据网络和水或天然气网络。在所有这些应用中，(G)NNP为多个非合作智能体的交互提供了强大的模型，这些非合作智能体优化了他们的策略。由于守恒律的解可能会发展不连续，它们表现出额外的非光滑现象，与对策相结合非常适合于SPP 1962的研究主题。基于最近关于守恒律网络解的存在性和稳定性以及守恒律的最优控制的结果，我们将发展一种分析环境，它给出了玩家代价泛函的稳定性和可微性。此外，我们还将得到一个基于伴随的导数表示。这将被用来研究非凸NNN的拟Nash平衡(QNE)的存在性以及这类非凸GNN的QNE和拟变分平衡(QVE)。对于具有凸可行集的对策，研究了QNE和QVE与基于正则化Nikaido-Isoda函数的价值函数的全局极小值的关系，并将其用于研究近似最佳响应图以证明存在结果。由于所考虑的对策是非凸的，我们计划建立这些价值函数的可微性，并针对凸约束(G)NEP发展全局收敛的下降方法。在非凸约束，特别是状态约束的情况下，将基于拉格朗日乘子和适当的约束限定导出QNE/QVE的概念，并将研究平衡点的存在性。对于具有非凸约束的(G)Net，我们将研究近似求解凸约束(G)Net序列的增广拉格朗日方法，上面的一类全局收敛方法可以应用于该序列。我们将探索通过非光滑牛顿步长加速这些下降方法的方法，以及通过块迭代进行分解的想法。虽然这些方法的灵感来自于由双曲网络管理的游戏的分析设置，但它们的设计也将涵盖其他受PDE限制的游戏。所开发的方法将在交通流和供应链模型中进行博弈实施和测试。

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 quasi-variational equilibria (QVE) for nonconvex GNEPs of this type. Here quasi-equilibria are characterized by variational inequalities that aggregate the players' first order optimality systems.For games with convex feasible sets, the relation of QNE and QVE 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 / QVE 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.