### Abstract

We study the problem of minimizing a sum of p-norms where p is a fixed real number in the interval [1, ∞]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the p-norm problem is not self-dual unless p = 2. Nevertheless, we are able to construct two logarithmically homogeneous self-concordant barrier functions for this problem. The barrier parameter of the first barrier function does not depend on p. The barrier parameter of the second barrier function increases with p. Using both barrier functions, we present a primal-dual potential reduction algorithm to compute an ∈-optimal solution in polynomial time that is independent of p. Computational experiences with a Matlab implementation are also reported.

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

Pages (from-to) | 551-579 |

Number of pages | 29 |

Journal | SIAM Journal on Optimization |

Volume | 10 |

Issue number | 2 |

State | Published - 1999 |

Externally published | Yes |

### Fingerprint

### Keywords

- Facilities location
- Minimizing a sum of norms
- Polynomial time algorithms
- Primal-dual potential reduction algorithms
- Shortest network under a given topology
- Steiner minimum trees

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Optimization*,

*10*(2), 551-579.

**An efficient algorithm for minimizing a sum of p-norms.** / Xue, Guoliang; Ye, Yinyu.

Research output: Contribution to journal › Article

*SIAM Journal on Optimization*, vol. 10, no. 2, pp. 551-579.

}

TY - JOUR

T1 - An efficient algorithm for minimizing a sum of p-norms

AU - Xue, Guoliang

AU - Ye, Yinyu

PY - 1999

Y1 - 1999

N2 - We study the problem of minimizing a sum of p-norms where p is a fixed real number in the interval [1, ∞]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the p-norm problem is not self-dual unless p = 2. Nevertheless, we are able to construct two logarithmically homogeneous self-concordant barrier functions for this problem. The barrier parameter of the first barrier function does not depend on p. The barrier parameter of the second barrier function increases with p. Using both barrier functions, we present a primal-dual potential reduction algorithm to compute an ∈-optimal solution in polynomial time that is independent of p. Computational experiences with a Matlab implementation are also reported.

AB - We study the problem of minimizing a sum of p-norms where p is a fixed real number in the interval [1, ∞]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the p-norm problem is not self-dual unless p = 2. Nevertheless, we are able to construct two logarithmically homogeneous self-concordant barrier functions for this problem. The barrier parameter of the first barrier function does not depend on p. The barrier parameter of the second barrier function increases with p. Using both barrier functions, we present a primal-dual potential reduction algorithm to compute an ∈-optimal solution in polynomial time that is independent of p. Computational experiences with a Matlab implementation are also reported.

KW - Facilities location

KW - Minimizing a sum of norms

KW - Polynomial time algorithms

KW - Primal-dual potential reduction algorithms

KW - Shortest network under a given topology

KW - Steiner minimum trees

UR - http://www.scopus.com/inward/record.url?scp=0011032427&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011032427&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0011032427

VL - 10

SP - 551

EP - 579

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 2

ER -