Weiszfeld's Method: Old and New Results

In 1937 the 16 years old Hungarian mathematician Endre Weiszfeld devised a method for solving the Fermat-Weber location problem – a problem whose origins can be traced back to the 17th century. This seminal paper of Weiszfeld [51] stirred up an enormous amount of research in the optimization and location communities, and is also being discussed and used till these days. In this paper we review both the past and the ongoing research on Weiszfed’s method. The existing results are presented in a self contained and concise manner – some are derived by new and simplified techniques. We also establish two new results using modern tools of optimization. First, we establish a nonasymptotic O(1/k) rate of convergence of Weiszfeld’s method, and second, using an exact smoothing technique, we present a modification of the method with a proven O(1/k2) rate of convergence.