Calculus of Variations and Geometric Measure Theory

A. Brancolini - G. Buttazzo - F. Santambrogio - E. Stepanov

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

created by stepanov on 26 Sep 2017


Published Paper

Inserted: 26 sep 2017

Journal: ESAIM Control Optim. Calc. Var.
Volume: 15
Pages: 509-524
Year: 2009
Doi: 10.1051/cocv:2008034

ArXiv: math/0612718 PDF
Links: Additional resources


Given the probability measure $\nu$ over the given region $\Omega\subset *R*^n$, we consider the optimal location of a set $\Sigma$ composed by $n$ points $\Omega$ in order to minimize the average distance $\Sigma\mapsto \int_\Omega {\rm dist}(x,\Sigma)\,d\nu$ (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.

Keywords: optimal transportation, average distance functional, location problem, facility location, Fermat-Weber problem, $k$-median problem, sequential allocation