### Abstract

In this paper, we study the problem of facility location on a sphere. This is a generalization of the planar Euclidean facility location problem. This problem was first studied by Katz and Cooper and by Drezner and Wesolowsky where Weiszfeld-like algorithms were proposed. However, convergence has never been proved. In this paper, we first prove a hull property of the problem, i.e., every global minimizer is in the spherical convex hull of the existing facilities. We then study the relationship between the spherical facility location problem and a planar Euclidean facility location problem corresponding to each approximate solution to the spherical facility location problem. Optimality conditions for the spherical facility location problem are established in terms of optimality conditions for the corresponding planar Euclidean facility location problem and a gradient algorithm is proposed to solve the spherical facility location problem. We prove that our algorithm always converges to a global minimizer of the spherical facility location problem. Computational results are also given.

Original language | English (US) |
---|---|

Pages (from-to) | 37-50 |

Number of pages | 14 |

Journal | Computers and Mathematics with Applications |

Volume | 27 |

Issue number | 6 |

DOIs | |

State | Published - 1994 |

Externally published | Yes |

### Fingerprint

### Keywords

- Global convergence
- Gradient algorithms
- Iterative algorithms without line search
- Open problems
- Optimality conditions
- Spherical facility location

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Modeling and Simulation

### Cite this

**A globally convergent algorithm for facility location on a sphere.** / Xue, Guoliang.

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 27, no. 6, pp. 37-50. https://doi.org/10.1016/0898-1221(94)90109-0

}

TY - JOUR

T1 - A globally convergent algorithm for facility location on a sphere

AU - Xue, Guoliang

PY - 1994

Y1 - 1994

N2 - In this paper, we study the problem of facility location on a sphere. This is a generalization of the planar Euclidean facility location problem. This problem was first studied by Katz and Cooper and by Drezner and Wesolowsky where Weiszfeld-like algorithms were proposed. However, convergence has never been proved. In this paper, we first prove a hull property of the problem, i.e., every global minimizer is in the spherical convex hull of the existing facilities. We then study the relationship between the spherical facility location problem and a planar Euclidean facility location problem corresponding to each approximate solution to the spherical facility location problem. Optimality conditions for the spherical facility location problem are established in terms of optimality conditions for the corresponding planar Euclidean facility location problem and a gradient algorithm is proposed to solve the spherical facility location problem. We prove that our algorithm always converges to a global minimizer of the spherical facility location problem. Computational results are also given.

AB - In this paper, we study the problem of facility location on a sphere. This is a generalization of the planar Euclidean facility location problem. This problem was first studied by Katz and Cooper and by Drezner and Wesolowsky where Weiszfeld-like algorithms were proposed. However, convergence has never been proved. In this paper, we first prove a hull property of the problem, i.e., every global minimizer is in the spherical convex hull of the existing facilities. We then study the relationship between the spherical facility location problem and a planar Euclidean facility location problem corresponding to each approximate solution to the spherical facility location problem. Optimality conditions for the spherical facility location problem are established in terms of optimality conditions for the corresponding planar Euclidean facility location problem and a gradient algorithm is proposed to solve the spherical facility location problem. We prove that our algorithm always converges to a global minimizer of the spherical facility location problem. Computational results are also given.

KW - Global convergence

KW - Gradient algorithms

KW - Iterative algorithms without line search

KW - Open problems

KW - Optimality conditions

KW - Spherical facility location

UR - http://www.scopus.com/inward/record.url?scp=38149144231&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149144231&partnerID=8YFLogxK

U2 - 10.1016/0898-1221(94)90109-0

DO - 10.1016/0898-1221(94)90109-0

M3 - Article

VL - 27

SP - 37

EP - 50

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 6

ER -