A globally convergent algorithm for facility location on a sphere

In this paper, we study the problem of facility location on a sphere. This is a generalization of the planar Euclidean facility location problem. This problem was first studied by Katz and Cooper and by Drezner and Wesolowsky where Weiszfeld-like algorithms were proposed. However, convergence has never been proved. In this paper, we first prove a hull property of the problem, i.e., every global minimizer is in the spherical convex hull of the existing facilities. We then study the relationship between the spherical facility location problem and a planar Euclidean facility location problem corresponding to each approximate solution to the spherical facility location problem. Optimality conditions for the spherical facility location problem are established in terms of optimality conditions for the corresponding planar Euclidean facility location problem and a gradient algorithm is proposed to solve the spherical facility location problem. We prove that our algorithm always converges to a global minimizer of the spherical facility location problem. Computational results are also given.