Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures

CentER Discussion Paper Series No. 2015-047

54 Pages Posted: 14 May 2014 Last revised: 29 Sep 2015

See all articles by Krzysztof Postek

Krzysztof Postek

Tilburg University - Center for Economic Research (CentER)

Dick den Hertog

Tilburg University - Department of Econometrics & Operations Research

Bertrand Melenberg

Tilburg University - Center for Economic Research (CentER)

Date Written: September 24, 2015

Abstract

In optimization problems appearing in fields such as economics, finance, or engineering, it is often important that a risk measure of a decision-dependent random variable stays below a prescribed level. At the same time, the underlying probability distribution determining the risk measure’s value is typically known only up to a certain degree and the constraint should hold for a reasonably wide class of probability distributions. In addition to that, the constraint should be computationally tractable. In this paper we review and generalize results on the derivation of tractable counterparts of such constraints for discrete probability distributions. Using established techniques in robust optimization, we show that the derivation of a tractable robust counterpart can be split into two parts: one corresponding to the risk measure and the other to the uncertainty set. This holds for a wide range of risk measures and uncertainty sets for probability distributions defined using statistical goodness-of-fit tests or probability metrics. In this way, we provide a unified framework of reformulating this class of constraints, extending the number of solvable risk measure-uncertainty set combinations considerably, including also risk measures that are nonlinear in the probabilities. To provide a clear overview for the user, we give the computational tractability status for each of the uncertainty set-risk measure pairs of which some have been solved in the literature. Examples, including portfolio optimization and antenna array design, illustrate the proposed approach in a theoretical and numerical setting.

Keywords: risk measure, robust counterpart, nonlinear inequality, robust optimization

JEL Classification: C61

Suggested Citation

Postek, Krzysztof and den Hertog, Dick and Melenberg, Bertrand, Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (September 24, 2015). CentER Discussion Paper Series No. 2015-047. Available at SSRN: https://ssrn.com/abstract=2435921 or http://dx.doi.org/10.2139/ssrn.2435921

Krzysztof Postek (Contact Author)

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands

Dick Den Hertog

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands

Bertrand Melenberg

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands
+31 13 466 2730 (Phone)

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