Solving Regularized Multifacility Location Problems with Unknown Number of Centers via Difference-of-Convex Optimization

TL;DR

This paper introduces a novel optimization approach for multifacility location problems with unknown centers, combining difference-of-convex techniques and smoothing methods to improve solution efficiency and applicability.

Contribution

It develops a new model using Minkowski gauge with Laplace regularization and proposes algorithms to determine the number of centers in location and clustering problems.

Findings

Proven existence of optimal solutions.

Effective algorithms for unknown center determination.

Extended recent multifacility location methodologies.

Abstract

In this paper, we develop optimization methods for a new model of multifacility location problems defined by a Minkowski gauge with Laplace-type regularization terms. The model is analyzed from both theoretical and numerical perspectives. In particular, we establish the existence of optimal solutions and study qualitative properties of global minimizers. By combining Nesterov's smoothing technique with recent advances in difference-of-convex optimization, following the pioneering work of P. D. Tao and L. T. H. An and others, we propose efficient numerical algorithms for minimizing the objective function of this model. As an application, our approach provides an effective method for determining the number of centers in gauge-based multifacility location and clustering problems. Our results extend and complement recent developments.