A projected Weiszfeld algorithm for the box-constrained Weber location problem

The Fermat-Weber problem consists in finding a point in R that minimizes the weighted sum of distances from m points in R that are not collinear. An application that motivated this problem is the optimal location of industries in the 2-dimensional case. The Weber problem is a generalization of the well-known Fermat problem. An usual method to solve the Weber problem, proposed by Weiszfeld in 1937, is based in a fixed-point iteration. In recent years there has been a growing interest in formalizing properties of well-definition of the method and proving convergence results. In this work we generalize the Weber location problem considering box-constrained restrictions. We propose a fixed-point iteration with projections on the restrictions and demonstrate descending properties. It is also proved that the limit of the sequence generated by the method is a feasible point and satisfies the KKT optimality conditions. Numerical experiments are presented to illustrate the theoretical results.