### Abstract

In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an ∈-optimal solution can be computed efficiently using interior-point algorithms. As applications to this problem, polynomial-time algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using Gaussian elimination on leaves of a tree, we present an algorithm which computes an ∈-optimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in O(N√N(log(c̄/∈) + log N)) arithmetic operations where c̄ is the largest pairwise distance among the given points. The previous best-known result on this problem is a graphical algorithm which requires O(N^{2}) arithmetic operations under certain conditions.

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

Pages (from-to) | 1017-1036 |

Number of pages | 20 |

Journal | SIAM Journal on Optimization |

Volume | 7 |

Issue number | 4 |

State | Published - Nov 1997 |

Externally published | Yes |

### Fingerprint

### Keywords

- Euclidean facilities location
- Interior-point algorithm
- Minimizing a sum of euclidean norms
- Polynomial time
- Shortest networks
- Steiner minimum trees

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Optimization*,

*7*(4), 1017-1036.

**An efficient algorithm for minimizing a sum of euclidean norms with applications.** / Xue, Guoliang; Ye, Yinyu.

Research output: Contribution to journal › Article

*SIAM Journal on Optimization*, vol. 7, no. 4, pp. 1017-1036.

}

TY - JOUR

T1 - An efficient algorithm for minimizing a sum of euclidean norms with applications

AU - Xue, Guoliang

AU - Ye, Yinyu

PY - 1997/11

Y1 - 1997/11

N2 - In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an ∈-optimal solution can be computed efficiently using interior-point algorithms. As applications to this problem, polynomial-time algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using Gaussian elimination on leaves of a tree, we present an algorithm which computes an ∈-optimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in O(N√N(log(c̄/∈) + log N)) arithmetic operations where c̄ is the largest pairwise distance among the given points. The previous best-known result on this problem is a graphical algorithm which requires O(N2) arithmetic operations under certain conditions.

AB - In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an ∈-optimal solution can be computed efficiently using interior-point algorithms. As applications to this problem, polynomial-time algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using Gaussian elimination on leaves of a tree, we present an algorithm which computes an ∈-optimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in O(N√N(log(c̄/∈) + log N)) arithmetic operations where c̄ is the largest pairwise distance among the given points. The previous best-known result on this problem is a graphical algorithm which requires O(N2) arithmetic operations under certain conditions.

KW - Euclidean facilities location

KW - Interior-point algorithm

KW - Minimizing a sum of euclidean norms

KW - Polynomial time

KW - Shortest networks

KW - Steiner minimum trees

UR - http://www.scopus.com/inward/record.url?scp=0031285688&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031285688&partnerID=8YFLogxK

M3 - Article

VL - 7

SP - 1017

EP - 1036

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 4

ER -