### Abstract

A proximity theorem is a statement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in a certain neighborhood of a solution to the relaxation. Proximity theorems have been used successfully, for example, in designing efficient algorithms for discrete resource allocation problems. After reviewing the recent results for L-convex and M-convex functions, this paper establishes proximity theorems for larger classes of discrete convex functions, L_{2}-convex functions and M _{2}-convex functions, that are relevant to the polymatroid intersection problem and the submodular flow problem.

Original language | English |
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Pages (from-to) | 539-562 |

Number of pages | 24 |

Journal | Mathematical Programming |

Volume | 99 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2004 Apr 1 |

Externally published | Yes |

### Keywords

- Discrete convex analysis
- Optimality criteria
- Proximity properties

### ASJC Scopus subject areas

- Software
- Mathematics(all)

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## Cite this

Murota, K., & Tamura, A. (2004). Proximity theorems of discrete convex functions.

*Mathematical Programming*,*99*(3), 539-562. https://doi.org/10.1007/s10107-003-0466-7