Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (revision of CentER DP 2014-031)

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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.
Original languageEnglish
Place of PublicationTilburg
PublisherOperations research
Number of pages54
Volume2015-047
Publication statusPublished - 28 Sep 2015

Publication series

NameCentER Discussion Paper
Volume2015-047

Fingerprint

Probability distributions
Finance
Antenna arrays
Random variables
Economics
Uncertainty

Keywords

  • risk measure
  • robust counterpart
  • nonlinear inequality
  • robust optimization
  • support functions

Cite this

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title = "Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (revision of CentER DP 2014-031)",
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.",
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Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (revision of CentER DP 2014-031). / Postek, K.S.; den Hertog, D.; Melenberg, B.

Tilburg : Operations research, 2015. (CentER Discussion Paper; Vol. 2015-047).

Research output: Working paperDiscussion paperOther research output

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