Data-Driven Methods for Decision-Making Under Uncertainty

Abstract

Robust optimization (RO) is a subfield of optimization which deals with uncertainty in the parameters of the program in the form of random variables. As opposed to stochastic optimization (SO) problems, which are in general non-tractable due to their probabilistic constraints, robust programs can be reformulated into tractable conic programs by characterizing the uncertain variables by an uncertainty set. The idea is to replace the probabilistic constraint of the random variable in stochastic programs by a constraint we wish to satisfy with high probability. This constraint is joined by an uncertainty set for which we estimate most instances of our random variable will lie. While some approaches exist to determine sets based on assumptions of the distributions of these random variables, these approaches are often not practical in cases where we have no prior knowledge of the underlying distribution of our data. By drawing on recent results in the closely related field of distributionally robust optimization (DRO), and more classical machine learning techniques of clustering, we present a data-driven method we coin mean robust optimization (MRO) for determining uncertainty sets and solving robust problems which exhibits attractive properties in both probabilistic guarantee and tractability.