On the Convergence of Miehle's Algorithm for the Euclidean Multifacility Location Problem

For the Euclidean single facility location problem, E. Weiszfeld proposed a simple iterative algorithm in 1937. Later, it was proved by numerous authors that it is a convergent descent algorithm. W. Miehle extended Weiszfeld's algorithm to solve the Euclidean multifacility location problem. Then, L. M. Ostresh proved that Miehle's algorithm is a descent algorithm. Recently, F. Rado modified Miehle's algorithm and provided several sets of sufficient conditions for the modified algorithm to converge. He also indicated that the convergence of Miehle's algorithm was an open problem. In this paper, the relationship between Miehle's multifacility location algorithm and Weiszfeld's single facility location algorithm is analyzed. Counterexamples show that Miehle's algorithm may converge to nonoptimal points for well structured problems.