An efficient algorithm for the Euclidean r-centrum location problem

In this paper we consider the single-facility Euclidean r-centrum location problem in Rn, which generalizes and unifies the classical 1-center and 1-median problem. Specifically, we reformulate this problem as a nonsmooth optimization problem only involving the maximum function, and then develop a smoothing algorithm that is shown to be globally convergent. The method transforms the original nonsmooth problem with certain combinatorial property into the solution of a deterministic smooth unconstrained optimization problem. Numerical results are presented for some problems generated randomly, indicating that the algorithm proposed here is extremely efficient for large problems.