ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

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ESAIM: COCV
DOI: 10.1051/cocv:2008034

Long-term planning versus short-term planning in the asymptotical location problem

Alessio Brancolini¹, Giuseppe Buttazzo², Filippo Santambrogio³ and Eugene Stepanov⁴

¹ SISSA, 4 via Beirut, 34014 Trieste, Italy; brancoli@sissa.it
² Dipartimento di Matematica, Università di Pisa, 5 Largo B. Pontecorvo, 56127 Pisa, Italy; buttazzo@dm.unipi.it
³ CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France; santambrogio@ceremade.dauphine.fr
⁴ Dipartimento di Matematica, Università di Pisa, 5 Largo B. Pontecorvo, 56127 Pisa, Italy; stepanov.eugene@gmail.com

Received February 28, 2007. Published online May 30, 2008.

Abstract
Given the probability measure $\nu$ over the given region $\Omega\subset \mathbb{R} ^n$ , we consider the optimal location of a set $\Sigma$ composed by n points in ${\Omega}$ in order to minimize the average distance $\Sigma\mapsto \int_{\Omega}\mathrm{dist}\,(x,\Sigma)\,{\rm d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving the configuration built at previous steps. We show that the respective optimization problems exhibit qualitatively different asymptotic behavior as $n \to \infty$ , although the optimization costs in both cases have the same asymptotic orders of vanishing.

Mathematics Subject Classification. 90B80, 90B85, 49J45, 46N10, 60K30

Key words: Location problem, facility location, Fermat-Weber problem, k-median problem, sequential allocation, average distance functional, optimal transportation

© EDP Sciences, SMAI 2008