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

该建议涉及随机拟变分不等式（QVI）约束的椭圆型和抛物型的优化问题。该项目的动机是应用程序的例子中，参数QVI内的随机变量，在许多情况下，他们的分布只有部分已知的真实的测量数据，或先验可能的情况。由于QVI是非凸的，和非光滑的问题（一般）多个解决方案，特殊的数学机器提出的随机解集的可测性和扰动分析的研究，并选择特定的解决方案。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，如特定数量的预期值，后者考虑到概率分布可能不精确。理论方面涉及的解决方案的存在性和扰动分析被认为是在同一时间的随机QVI和整体优化问题的解决方案算法提出。这进一步包括离散化方法的计划，这些方法旨在良性场景中打破维度灾难。

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.