Revisiting several problems and algorithms in continuous location with $$\ell _\tau $$ ℓ τ norms

This paper addresses the general continuous single facility location problems in finite dimension spaces under possibly different $$\ell _\tau $$ ℓ τ norms, $$\tau \ge 1$$ τ ≥ 1 , in the demand points. We analyze the difficulty of this family of problems and revisit convergence properties of some well-known algorithms. The ultimate goal is to provide a common approach to solve the family of continuous $$\ell _\tau $$ ℓ τ ordered median location problems Nickel and Puerto (Facility location: a unified approach, 2005 ) in dimension $$d$$ d (including of course the $$\ell _\tau $$ ℓ τ minisum or Fermat-Weber location problem for any $$\tau \ge 1$$ τ ≥ 1 ). We prove that this approach has a polynomial worst case complexity for monotone lambda weights and can be also applied to constrained and even non-convex problems. Copyright Springer Science+Business Media New York 2014