Generalized Nash Equilibrium Problems with Partial Differential Operators: Theory, Algorithms, and Risk Aversion

本文研究不确定条件下具有偏微分方程约束的广义纳什均衡问题及其在函数空间中的推广。后者包括具有平衡约束的平衡问题（EPECs）和无限维的具有平衡约束的多重优化问题（MOPECs）。考虑的pde约束问题的新类别提供了广泛的数学新颖性。从理论角度来看，存在性和平稳性结果的推导需要非光滑优化和集值分析的技术，包括不动点定理和集值映射的广义微分。另一方面，这些问题的变分形式通常缺乏必要的单调性，这将允许从研究非光滑变分问题和标准pde约束优化中直接应用已知的基于函数空间的数值方法。提出的框架扩展了经典的数学规划范式，超越了单目标优化问题，因此，允许人们考虑科学和经济学中更广泛的平衡问题。特别是，具有层次解决概念的问题，如纳什均衡或纳什-斯塔克尔伯格均衡（多领导-追随者均衡）可以被考虑。平稳性条件的存在性证明和推导将与快速数值算法的推导一起进行，这些算法考虑了偏微分算子和分布参数设置的微妙之处。为了在具有不确定性的复杂系统的竞争环境中建立风险厌恶模型，采用了一致的风险度量。提出了一个四部分的研究计划，以严格发展存在性和平稳性理论，算法和数值分析，并通过风险措施处理不确定性。

The subject of this proposal is generalized Nash equilibrium problems (GNEPs) with partial differential equation (PDE) constraints under uncertainty and their extension to new classes of equilibrium problems in function space. The latter includes so-called equilibrium problems with equilibrium constraints (EPECs) and multiple optimization problems with equilibrium constraints (MOPECs) in infinite dimensions. The new classes of PDE-constrained problems under consideration offer a wide range of mathematical novelty. From a theoretical perspective, the derivation of existence and stationarity results requires techniques from non-smooth optimization and set-valued analysis, which includes fixed point theorems and generalized differentiation for set-valued mappings. On the other hand, the variational form of these problems typically lack the necessary monotonicity properties that would allow the direct application of known function-space based numerical methods from the study of non-smooth variational problems and standard PDE-constrained optimization. The proposed framework extends the classical mathematical programming paradigm beyond a single-objective optimization problem and thus, allows one to consider a broader array of equilibrium problems in the sciences and economics. In particular, problems with hierarchical solution concepts such as Nash equilibria or Nash-Stackelberg equilibria (multi-leader-follower equilibria) can be considered. The existence proofs and derivation of stationarity conditions are to be carried out in conjunction with the derivation of fast numerical algorithms, which take into account the subtleties of partial differential operators and the distributed parameter setting. In order to model risk-aversion in a competitive setting with a complex system subject to uncertainties, coherent risk measures are employed. A four part research program is proposed for the rigorous development of an existence and stationarity theory, algorithms and numerical analysis, and handling uncertainties via risk measures.