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

偏微分算子的广义纳什均衡问题：理论、算法和风险规避

• 批准号: 314141981
• 负责人: Professor Dr. Michael Hintermüller
• 金额: --
• 依托单位: Forschungsgruppe Nichtglatte Variationsprobleme und Operatorgleichungen
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314141981?language=en
• 关键词: Generalized; Nash; Equilibrium; Problems; Partial

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.

这个建议的主题是广义纳什均衡问题（GNEPs）的偏微分方程（PDE）的约束下的不确定性和它们的扩展到新的类的平衡问题的功能空间。后者包括所谓的平衡问题与平衡约束（EPEC）和多重优化问题与平衡约束（MOPEC）在无限维。考虑中的PDE约束问题的新类别提供了广泛的数学新奇。从理论的角度来看，存在性和平稳性结果的推导需要来自非光滑优化和集值分析的技术，其中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属性，这将允许直接应用已知的基于函数空间的数值方法从非光滑变分问题和标准PDE约束优化的研究。 所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，允许人们考虑更广泛的平衡问题，在科学和经济学。特别是，可以考虑具有分层解决方案概念的问题，例如纳什均衡或纳什-斯塔克伯格均衡（多领导者-追随者均衡）。稳定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，其中考虑到偏微分算子和分布参数设置的微妙之处。 为了模拟风险规避在竞争环境中的复杂系统的不确定性，相干风险措施。一个由四部分组成的研究计划，提出了严格的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。