This paper considers the Fermat-Weber location problem. It is shown that, after a suitable initialization, the standard Newton method can be applied to the Fermat-Weber problem and is globally and locally quadratically convergent. A numerical comparison with the popular Weiszfeld algorithm shows that Newton’s method is significantly more efficient than the Weiszfeld scheme.

In 1937 the 16 years old Hungarian mathematician Endre Weiszfeld devised a method for solving the Fermat-Weber location problem – a problem whose origins can be traced back to the 17th century. This seminal paper of Weiszfeld [51] stirred up an enormous amount of research in the optimization and location communities, and is also being discussed and used till these days. In this paper we review ...

MINIMIZING THE SUM OF EUCLIDEAN DISTANCES YUYING LI Abstract. The Weiszfeld algorithm for continuous location problems can be considered as an iteratively reweighted least squares method. It exhibits linear convergence. In this paper, a Newton type algorithm with similar simplicity is proposed to solve a continuous multifacility location problem with Euclidean distance measure. Similar to the W...

A new accelerated algorithm to solve the single-facility minisum location problem is developed. The acceleration is achieved using a feedback factor. The proposed algorithm converges faster than the accelerating procedures available in the literature. Being nearly as simple as the classical Weiszfeld procedure, the new method can easily be implemented in real applications. Practical subroutines...

The Fermat–Weber location problem (also single facility location problem) is to locate a facility that will serve optimally a set of customers, given by their locations and weights, in the sense of minimizing the weighted sum of distances traveled by the customers. A well known method for solving the problem is the Weiszfeld method [25], a gradient method that expresses and updates the sought c...

This paper presents a method for finding an Lqclosest-point to a set of affine subspaces, that is a point for which the sum of the q-th power of orthogonal distances to all the subspaces is minimized, where 1 ≤ q < 2. We give a theoretical proof for the convergence of the proposed algorithm to a unique Lq minimum. The proposed method is motivated by the Lq Weiszfeld algorithm, an extremely simp...

We present a new iterative method for probabilistic clustering of data. Given clusters, their centers and the distances of data points from these centers, the probability of cluster membership at any point is assumed inversely proportional to the distance from (the center of) the cluster in question. This assumption is our working principle. The method is a generalization, to several centers, o...