Theory and applications of Distributionally Robust Optimization with side data

Abstract

Nowadays, a large amount of varied data is being generated which, when made available to the decision maker, constitutes a valuable resource in optimization problems. These data, however, are not free from uncertainty about the physical, economic or social context, system or process from which they originate; uncertainty that, on the other hand, the decision maker must take into account in his/her decision making process. The objective of this PhD dissertation is to develop theoretical foundations and investigate methods for solving optimization problems where there is a great diversity of data on uncertain phenomena. Today’s decision makers not only collect observations from the uncertainties directly affecting their decision-making processes, but also gather some prior information about the data-generating distribution of the uncertainty. This information is used by the decision maker to prescribe a more accurate set of potential probability distributions, the so-called ambiguity set in distributionally robust optimization. Our intention, therefore, is to develop a purely data-driven methodology, within the scope of distributionally robust optimization based on the optimal transportation problem, which exploits some extra/prior information about the random phenomenon. This extra information crystallizes in two axes on the nature of the random phenomenon: first, some prior information about, for example, the shape/structure of the probability distribution; second, some conditional information such as that given by various covariates, which help explain the random phenomenon underlying the optimization problem without resorting to prior regression techniques.