### Abstract

In discrete convex analysis, the scaling and proximity properties for the class of L(Formula presented.)-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when (Formula presented.), while a proximity theorem can be established for any n, but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.

Original language | English |
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Pages (from-to) | 1-36 |

Number of pages | 36 |

Journal | Mathematical Programming |

DOIs | |

Publication status | Accepted/In press - 2018 Jan 24 |

### ASJC Scopus subject areas

- Software
- Mathematics(all)

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## Cite this

*Mathematical Programming*, 1-36. https://doi.org/10.1007/s10107-018-1234-z