Local solutions of the multi-source Weber problem

Several new qualitative properties of the problem of minimizing the sum of the weighted minima of the Euclidean distances of the demand points to the facilities, which is called the multi-source Weber problem and also known as the clustering problem with Euclidean norms, are obtained in this paper. Nontrivial local solutions are defined and characterized with the help of a necessary optimality condition in DC programming and the special structure of the problem in question. Since most of the existing solution algorithms for this problem just give some local solutions, our characterizations can be used to analyse and refine these algorithms.