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

Guoliang Xue, Yinyu Ye

Research output: Contribution to journalArticle

72 Scopus citations

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(N2) arithmetic operations under certain conditions.

Original languageEnglish (US)
Pages (from-to)1017-1036
Number of pages20
JournalSIAM Journal on Optimization
Volume7
Issue number4
DOIs
StatePublished - Nov 1997

Keywords

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

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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