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

该项目的目的是分析（广义）纳什均衡问题（（G）NEPs），这些问题由非线性双曲守恒或平衡律网络控制，以及开发和分析这些问题的有效解决方法。守恒定律网络是一个活跃的研究领域，并导致了流动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）NEP为优化其策略的多个非合作代理的交互提供了强大的模型。由于守恒律的解可能产生不连续性，它们表现出额外的非光滑现象，这些现象与博弈相结合，完全符合SPP 1962的研究主题。基于最近的结果，关于网络的守恒律以及守恒律的最优控制的解决方案的存在性和稳定性，我们将开发一个分析设置，产生稳定性和可微性能的球员的成本泛函。此外，我们将推导出一个基于伴随的导数表示。这将被用来研究非凸NEP的准纳什均衡（QNE）的存在性，以及这种类型的非凸GNEP的QNE和准变分均衡（QVE）。本文利用变分不等式来刻画拟均衡，并对具有凸可行集的对策，研究了QNE和QVE与基于正则化Nikaido-Isoda函数的价值函数全局极小值的关系，并将其用于研究最佳对策映射的存在性.由于所考虑的游戏是非凸的，我们计划建立这些价值函数的可微性，并开发凸约束（G）NEP的全局收敛下降方法。在非凸约束的情况下，特别是状态约束，QNE / QVE概念的基础上拉格朗日乘子和适当的约束资格将被导出和平衡的存在性进行研究。对于具有非凸约束的（G）NEP，我们将研究近似求解凸约束（G）NEP序列的增广拉格朗日方法，上述一类全局收敛的方法可以应用于该方法。将探讨通过非光滑牛顿步骤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双曲网络游戏的分析设置，但它们也将被设计为涵盖其他PDE约束游戏。所开发的方法将实施和测试的交通流量和供应链模型的游戏。

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.