### Abstract

In this paper we study distributionally robust constraints on risk measures (such

as standard deviation less the mean, Conditional Value-at-Risk, Entropic Value-at-Risk) of decision-dependent random variables. The uncertainty sets for the discrete probability distributions are defined using statistical goodness-of-fit tests and probability metrics such as Pearson, likelihood ratio, Anderson-Darling tests, or Wasserstein distance. This type of constraints arises in problems in portfolio optimization, economics, machine learning, and engineering. 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. We also show how the counterpart can be constructed for risk measures that are nonlinear in the probabilities (for example, variance or the Conditional Value-at-Risk). We provide the computational tractability status for each of the uncertainty set-risk measure pairs that we could solve. Numerical examples including portfolio optimization and a multi-item newsvendor problem illustrate the proposed approach.

as standard deviation less the mean, Conditional Value-at-Risk, Entropic Value-at-Risk) of decision-dependent random variables. The uncertainty sets for the discrete probability distributions are defined using statistical goodness-of-fit tests and probability metrics such as Pearson, likelihood ratio, Anderson-Darling tests, or Wasserstein distance. This type of constraints arises in problems in portfolio optimization, economics, machine learning, and engineering. 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. We also show how the counterpart can be constructed for risk measures that are nonlinear in the probabilities (for example, variance or the Conditional Value-at-Risk). We provide the computational tractability status for each of the uncertainty set-risk measure pairs that we could solve. Numerical examples including portfolio optimization and a multi-item newsvendor problem illustrate the proposed approach.

Original language | English |
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Place of Publication | Tilburg |

Publisher | Operations research |

Number of pages | 39 |

Volume | 2014-031 |

Publication status | Published - 9 May 2014 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2014-031 |

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### Keywords

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

### Cite this

Postek, K. S., den Hertog, D., & Melenberg, B. (2014).

*Tractable Counterparts of Distributionally Robust Constraints on Risk Measures*. (CentER Discussion Paper; Vol. 2014-031). Tilburg: Operations research.