Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with lp Distances

This paper considers a general form of the single facility minisum location problem (also referred to as the Fermat-Weber problem), where distances are measured by an l_p norm. An iterative solution algorithm is given which generalizes the well-known Weiszfeld procedure for Euclidean distances. Global convergence of the algorithm is proven for any value of the parameter p in the closed interval [1, 2], provided an iterate does not coincide with a singular point of the iteration functions. However, for p > 2, the descent property of the algorithm and as a result, global convergence, are no longer guaranteed. These results generalize the work of Kuhn for Euclidean (p = 2) distances.