A Newton Acceleration of the Weiszfeld Algorithm forMinimizing the Sum of Euclidean Distances

  • Authors:

  • Yuying Li

  • Affiliations:

  • Department of Computer Science, Cornell University, Ithaca, NY 14850. E-mail: yuying@cs.cornell.edu

  • Venue:
  • Computational Optimization and Applications
  • Year:
  • 1998

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Abstract

The Weiszfeld algorithm for continuous location problems can beconsidered as an iteratively reweighted least squares method. It generallyexhibits linear convergence. In this paper, a Newton algorithm with similarsimplicity is proposed to solve a continuous multifacility location problemwith the Euclidean distance measure. Similar to the Weiszfeld algorithm, themain computation can be solving a weighted least squares problem at eachiteration. A Cholesky factorization of a symmetric positive definite bandmatrix, typically with a small band width (e.g., a band width of two for aEuclidean location problem on a plane) is performed. This new algorithm canbe regarded as a Newton acceleration to the Weiszfeld algorithm with fastglobal and local convergence. The simplicity and efficiency of the proposedalgorithm makes it particularly suitable for large-scale Euclidean locationproblems and parallel implementation. Computational experience suggests thatthe proposed algorithm often performs well in the absence of the linearindependence or strict complementarity assumption. In addition, the proposedalgorithm is proven to be globally convergent under similar assumptions forthe Weiszfeld algorithm. Although local convergence analysis is still underinvestigation, computation results suggest that it is typicallysuperlinearly convergent.