A Unified Approach to Optimal Uncertainty Quantification and Risk-Averse Optimization withQuasi-Variational Inequality Constraints

具有拟变分不等式约束的最优不确定性量化和风险规避优化的统一方法

• 批准号: 423760521
• 负责人: Professor Dr. Michael Hintermüller, since 12/2019
• 金额: --
• 依托单位: Department of Mathematical Sciences
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/423760521?language=en
• 关键词: Unified; Approach; Optimal; Uncertainty; Quantification

This proposal concerns optimization problems with random quasi-variational inequality (QVI) constraints of elliptic and parabolic type. The project is motivated by examples of application where parameters within QVIs are random variables and in many occasions their distribution is only partially known either by real measured data, or a priori possible scenarios. Since QVIs are non-convex, and non-smooth problems with (in general) multiple solutions, special mathematical machinery is proposed for the study of measurability and perturbation analysis of the random solution set, and the selection of particular solutions thereof. The optimization problem class consider in a unified fashion risk-averse optimization and optimal uncertainty quantification: While the former deals with measures of risk like the expected value of a certain quantity of interest, the latter takes into account that probability distributions may not be known exactly. Theoretical aspects involving existence of solutions and perturbation analysis are considered and at the same time solution algorithms for the random QVIs and the overall optimization problems are proposed. This further includes plans of discretization approaches that aim in benign scenarios to break the curse of dimensionality.

该提案涉及椭圆形和抛物线类型的随机准变量不平等（QVI）约束的优化问题。该项目是由QVI中的参数为随机变量的应用程序的示例，在许多情况下，它们的分布仅通过实际测量数据或先验可能的方案而部分地知道。由于QVI是非凸，并且（通常）多个解决方案的非平滑问题，因此提出了特殊的数学机制，用于研究随机溶液集的可测量性和扰动分析，并选择了其特定溶液。优化问题类别以统一的时尚风险优化和最佳的不确定性量化考虑：虽然前者涉及风险度量，例如一定量的兴趣的预期价值，但后者考虑到概率分布可能并不确切知道。考虑了涉及解决方案和扰动分析的理论方面，同时提出了随机QVI的解决方案算法，并提出了整体优化问题。这进一步包括旨在在良性场景中打破维度诅咒的离散化方法的计划。