A D.C. Algorithm via Convex Analysis Approach for Solving a Location Problem Involving Sets

TL;DR

This paper introduces a new algorithm for solving a convex location problem involving weighted distances to convex sets, including negative weights, by combining d.c. programming and a generalized Weiszfeld algorithm.

Contribution

It presents a novel algorithm that extends existing methods to handle location problems with negative weights using convex analysis and d.c. programming techniques.

Findings

Algorithm effectively solves the problem with negative weights

Proves existence of solutions under certain conditions

Demonstrates practical applicability through examples

Abstract

We study a location problem that involves a weighted sum of distances to closed convex sets. As several of the weights might be negative, traditional solution methods of convex optimization are not applicable. After obtaining some existence theorems, we introduce a simple, but effective, algorithm for solving the problem. Our method is based on the Pham Dinh - Le Thi algorithm for d.c. programming and a generalized version of the Weiszfeld algorithm, which works well for convex location problems.