### 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 |
---|---|

Pages (from-to) | 539-562 |

Number of pages | 24 |

Journal | Mathematical Programming |

Volume | 99 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2004 Apr |

Externally published | Yes |

### Fingerprint

### Keywords

- Discrete convex analysis
- Optimality criteria
- Proximity properties

### ASJC Scopus subject areas

- Applied Mathematics
- Mathematics(all)
- Safety, Risk, Reliability and Quality
- Management Science and Operations Research
- Software
- Computer Graphics and Computer-Aided Design
- Computer Science(all)

### Cite this

*Mathematical Programming*,

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

**Proximity theorems of discrete convex functions.** / Murota, Kazuo; Tamura, Akihisa.

Research output: Contribution to journal › Article

*Mathematical Programming*, vol. 99, no. 3, pp. 539-562. https://doi.org/10.1007/s10107-003-0466-7

}

TY - JOUR

T1 - Proximity theorems of discrete convex functions

AU - Murota, Kazuo

AU - Tamura, Akihisa

PY - 2004/4

Y1 - 2004/4

N2 - 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, L2-convex functions and M 2-convex functions, that are relevant to the polymatroid intersection problem and the submodular flow problem.

AB - 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, L2-convex functions and M 2-convex functions, that are relevant to the polymatroid intersection problem and the submodular flow problem.

KW - Discrete convex analysis

KW - Optimality criteria

KW - Proximity properties

UR - http://www.scopus.com/inward/record.url?scp=21144451016&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21144451016&partnerID=8YFLogxK

U2 - 10.1007/s10107-003-0466-7

DO - 10.1007/s10107-003-0466-7

M3 - Article

VL - 99

SP - 539

EP - 562

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 3

ER -