Survey and Review

Abstract

The Survey and Review article in this issue is “Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures,” by Krzysztof Postek, Dick den Hertog, and Bertrand Melenberg. Readers will be aware of the growing interest in uncertainty quantification (UQ). Researchers in this field take account of modeling, observational, and computational errors and express these “solutions” in terms of distributions or confidence intervals. The topic of this article could be regarded as an example of what happens after UQ. The authors consider the case where the reward arising from a decision has many possible outcomes, represented by a discrete-valued random variable. For example, the decision may be to choose a convex combination of assets in order to construct a financial portfolio. Having quantified the uncertainty, we can define the associated risk. Then the problem of maximizing the expected return subject to an acceptable level of risk leads to a deterministic optimization problem. Robust optimization assumes that the parameters in the problem have known, characterized levels of uncertainty defined by ambiguity or uncertainty sets. This brings together ideas from numerical analysis, probability theory, and applied analysis and has many applications in science and engineering. The aim of the article is to provide a unified framework for setting up and analyzing problems of this type, for a wide range of risk measures and uncertainty sets. The emphasis is on massaging the problem into one that is computationally tractable. A building-block approach allows the authors to capture and extend many results from the literature---Table 1 gives a comprehensive summary of the cases that are covered. As well as adding concrete examples from portfolio optimization and antenna design, the authors point to some open questions and opportunities for related work. Overall, this Survey and Review article clarifies and supplements a well-defined and active field at the intersection between optimization and computational statistics.