### Abstract

For the Euclidean single facility location problem, E. Weiszfeld proposed a simple closed-form iterative algorithm in 1937. Later, numerous authors proved that it is a convergent descent algorithm. In 1973, J. Eyster, J. White and W. Wierwille extended Weiszfeld's idea and proposed a Hyperbloid. Approximation Procedure (HAP) for solving the Euclidean multifacility location problem. They believed, based on considerable computational experience, that the HAP always converges. In 1977, Ostresh proved that the HAP is a descent algorithm under certain conditions. In 1981, Morris proved that a variant of the HAP always converges. However, no convergence proof for the original HAP has ever been given. In this paper, we prove that the HAP is a descent algorithm and that it always converges to the minimizer of the objective function from any initial point.

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

Pages (from-to) | 1164-1171 |

Number of pages | 8 |

Journal | Operations Research |

Volume | 41 |

Issue number | 6 |

State | Published - Nov 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Management Science and Operations Research

### Cite this

*Operations Research*,

*41*(6), 1164-1171.

**On the convergence of a hyperboloid approximation procedure for the perturbed euclidean multifacility location problem.** / Rosen, J. B.; Xue, Guoliang.

Research output: Contribution to journal › Article

*Operations Research*, vol. 41, no. 6, pp. 1164-1171.

}

TY - JOUR

T1 - On the convergence of a hyperboloid approximation procedure for the perturbed euclidean multifacility location problem

AU - Rosen, J. B.

AU - Xue, Guoliang

PY - 1993/11

Y1 - 1993/11

N2 - For the Euclidean single facility location problem, E. Weiszfeld proposed a simple closed-form iterative algorithm in 1937. Later, numerous authors proved that it is a convergent descent algorithm. In 1973, J. Eyster, J. White and W. Wierwille extended Weiszfeld's idea and proposed a Hyperbloid. Approximation Procedure (HAP) for solving the Euclidean multifacility location problem. They believed, based on considerable computational experience, that the HAP always converges. In 1977, Ostresh proved that the HAP is a descent algorithm under certain conditions. In 1981, Morris proved that a variant of the HAP always converges. However, no convergence proof for the original HAP has ever been given. In this paper, we prove that the HAP is a descent algorithm and that it always converges to the minimizer of the objective function from any initial point.

AB - For the Euclidean single facility location problem, E. Weiszfeld proposed a simple closed-form iterative algorithm in 1937. Later, numerous authors proved that it is a convergent descent algorithm. In 1973, J. Eyster, J. White and W. Wierwille extended Weiszfeld's idea and proposed a Hyperbloid. Approximation Procedure (HAP) for solving the Euclidean multifacility location problem. They believed, based on considerable computational experience, that the HAP always converges. In 1977, Ostresh proved that the HAP is a descent algorithm under certain conditions. In 1981, Morris proved that a variant of the HAP always converges. However, no convergence proof for the original HAP has ever been given. In this paper, we prove that the HAP is a descent algorithm and that it always converges to the minimizer of the objective function from any initial point.

UR - http://www.scopus.com/inward/record.url?scp=0027703251&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027703251&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0027703251

VL - 41

SP - 1164

EP - 1171

JO - Operations Research

JF - Operations Research

SN - 0030-364X

IS - 6

ER -