### Abstract

Original language | English |
---|---|

Place of Publication | Tilburg |

Publisher | Operations research |

Number of pages | 54 |

Volume | 2015-047 |

Publication status | Published - 28 Sep 2015 |

### Publication series

Name | CentER Discussion Paper |
---|---|

Volume | 2015-047 |

### Fingerprint

### Keywords

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

### Cite this

*Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (revision of CentER DP 2014-031)*. (CentER Discussion Paper; Vol. 2015-047). Tilburg: Operations research.

}

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

Research output: Working paper › Discussion paper › Other research output

TY - UNPB

T1 - Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (revision of CentER DP 2014-031)

AU - Postek, K.S.

AU - den Hertog, D.

AU - Melenberg, B.

PY - 2015/9/28

Y1 - 2015/9/28

N2 - 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.

AB - 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.

KW - risk measure

KW - robust counterpart

KW - nonlinear inequality

KW - robust optimization

KW - support functions

M3 - Discussion paper

VL - 2015-047

T3 - CentER Discussion Paper

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

PB - Operations research

CY - Tilburg

ER -