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

这一建议涉及椭圆型和抛物型随机拟变分不等式(QVI)约束的优化问题。该项目的动机是应用实例，其中QVI中的参数是随机变量，并且在许多情况下，它们的分布仅通过实际测量数据或先验可能的情景而部分知道。由于QVI是具有(一般)多解的非凸、非光滑问题，为了研究随机解集的可测性、摄动分析及其特解的选择，提出了特殊的数学机制。优化问题类统一考虑风险厌恶优化和最优不确定性量化：前者处理风险度量，如一定数量的兴趣的期望值，而后者考虑概率分布可能不是精确知道的。从理论上考虑了解的存在性和摄动分析，同时给出了随机QVI问题和全局优化问题的求解算法。这还包括以良性情景为目标的离散化方法计划，以打破维度诅咒。

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.