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

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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.