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

The Weber problem consists of finding a point in Rn that minimizes the weighted sum of distances from m points in Rn that are not collinear. An application that motivated this problem is the optimal location of facilities in the 2-dimensional case. A classical method to solve the Weber problem, proposed by Weiszfeld in 1937, is based on a fixed-point iteration.In this work we generalize the Weber location problem considering box constraints. We propose a fixed-point iteration with projections on the constraints 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 validate the theoretical results.