A note on convergence in the single facility minisum location problem

The single facility minisum location problem requires finding a point in R-N that minimizes a sum of weighted distances to m given points. The distance measure is typically assumed in the literature to be either Euclidean or rectangular, or the more general l(p) norm. Global convergence of a well-known iterative solution method named the Weiszfeld procedure has been proven under the proviso that none of the iterates coincide with a singular point of the iteration functions. The purpose of this paper is to examine the corresponding set of "bad" starting points which result in failure of the algorithm for a general l(p) norm. An important outcome of this analysis is that the set of bad starting points will always have a measure zero in the solution space (RN), thereby validating the global convergence properties of the Weiszfeld procedure for any l(p) norm, p is an element of [1, 2].