## Abstract

We study stochastic programs where the decision maker cannot observe the distribution of the exogenous uncertainties but has access to a finite set of independent samples from this distribution. In this setting, the goal is to find a procedure that transforms the data to an estimate of the expected cost function under the unknown data-generating distribution, that is, a predictor, and an optimizer of the estimated cost function that serves as a near-optimal candidate decision, that is, a prescriptor. As functions of the data, predictors and prescriptors constitute statistical estimators. We propose a meta-optimization problem to find the least conservative predictors and prescriptors subject to constraints on their out-of-sample disappointment. The out-of-sample disappointment quantifies the probability that the actual expected cost of the candidate decision under the unknown true distribution exceeds its predicted cost. Leveraging tools from large deviations theory, we prove that this meta-optimization problem admits a unique solution: The best predictor-prescriptor-pair is obtained by solving a distributionally robust optimization problem over all distributions within a given relative entropy distance from the empirical distribution of the data.

Original language | English |
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Pages (from-to) | 3387-3402 |

Journal | Management Science |

Volume | 67 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2021 |

### Bibliographical note

Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.## Keywords

- Convex optimization
- Data-driven optimization
- Distributionally robust optimization
- Large deviations theory
- Relative entropy