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

研究椭圆型和抛物型随机拟变分不等式约束下的优化问题。该项目的动机来自于一些应用实例，其中QVIs中的参数是随机变量，并且在许多情况下，它们的分布只能通过实际测量数据或先验的可能场景部分地知道。由于QVIs是具有（一般）多解的非凸非光滑问题，提出了研究随机解集的可测性、摄动分析及其特解选择的特殊数学机制。优化问题类以统一的方式考虑风险规避优化和最优不确定性量化：前者处理风险度量，如一定数量的利息的期望值，后者考虑到概率分布可能不完全已知。考虑了解的存在性和摄动分析等理论问题，同时提出了随机qvi和整体优化问题的求解算法。这进一步包括离散化方法的计划，旨在在良性场景中打破维度的诅咒。

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.