### Abstract

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

Pages (from-to) | 603-650 |

Number of pages | 48 |

Journal | SIAM Review |

Volume | 58 |

Issue number | 4 |

DOIs | |

Publication status | Published - 3 Nov 2016 |

### Fingerprint

### Keywords

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

### Cite this

}

*SIAM Review*, vol. 58, no. 4, pp. 603-650. https://doi.org/10.1137/151005221

**Computationally tractable counterparts of distributionally robust constraints on risk measures.** / Postek, Krzysztof; den Hertog, Dick; Melenberg, Bertrand.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Computationally tractable counterparts of distributionally robust constraints on risk measures

AU - Postek, Krzysztof

AU - den Hertog, Dick

AU - Melenberg, Bertrand

PY - 2016/11/3

Y1 - 2016/11/3

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, 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 for reformulating this class of constraints, extending the number of solvable risk measure-uncertainty set combinations considerably, also including risk measures that are nonlinear in the probabilities. To provide a clear overview for the user, we provide the computational tractability status for each of the uncertainty set-risk measure pairs, some of which 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, 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 for reformulating this class of constraints, extending the number of solvable risk measure-uncertainty set combinations considerably, also including risk measures that are nonlinear in the probabilities. To provide a clear overview for the user, we provide the computational tractability status for each of the uncertainty set-risk measure pairs, some of which 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

U2 - 10.1137/151005221

DO - 10.1137/151005221

M3 - Article

VL - 58

SP - 603

EP - 650

JO - SIAM Review

JF - SIAM Review

SN - 0036-1445

IS - 4

ER -