## Abstract

We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that these problems cannot be approximated to any degree of either relative or additive error, regardless of the scheme used. To circumvent these hardness results, we generalize the concept of fully polynomial-time approximation scheme allowing arbitrarily small additive and multiplicative error at the same time, while requiring a polynomial running time in the input size and the error parameters. We develop approximation schemes of this type for the classes of problems mentioned above. In light of our hardness results, such approximation schemes are “best possible”. A computational evaluation shows the promise of this approach.

Original language | English |
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Pages (from-to) | 183-242 |

Number of pages | 60 |

Journal | Mathematical Programming |

Volume | 195 |

Issue number | 1-2 |

DOIs | |

State | Published - Sep 2022 |

### Bibliographical note

Publisher Copyright:© 2021, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

### Funding

Nir Halman: Work on this paper was done while the author was a faculty member at the Hebrew University of Jerusalem, Israel. Supported in part by the Israel Science Foundation, Grant Number 399/17 and the United States-Israel Binational Science Foundation, Grant Number 2018095. Giacomo Nannicini: Supported in part by the United States-Israel Binational Science Foundation, Grant Number 2018095.

Funders | Funder number |
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United States-Israel Binational Science Foundation | 2018095 |

Israel Science Foundation | 399/17 |

## Keywords

- Approximation algorithms
- Hardness of approximation
- K-approximation sets and functions
- Newsvendor problem
- Stochastic dynamic programming
- Stochastic inventory control