Minimizing Differences of Convex Functions and Applications to Facility Location and Clustering

TL;DR

This paper introduces algorithms for solving complex facility location and clustering problems involving convex functions, weights, and Minkowski distances, utilizing Nesterov smoothing and DCA methods.

Contribution

It presents novel algorithms for generalized Fermat-Torricelli, multifacility location, and a new convex set clustering model using advanced optimization techniques.

Findings

Effective algorithms for weighted Fermat-Torricelli problems.

Solutions for multifacility location with Minkowski distances.

A new convex set clustering model.

Abstract

In this paper we develop algorithms to solve generalized weighted Fermat-Torricelli problems with positive and negative weights and multifacility location problems involving distances generated by Minkowski gauges. We also introduce a new model of clustering based on squared distances to convex sets. Using the Nesterov smoothing technique and an algorithm for minimizing differences of convex functions called the DCA introduced by Tao and An, we develop effective algorithms for solving these problems.