Long-term planning versus short-term planning in the asymptotical location problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 509-524.

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 in $\Omega$ in order to minimize the average distance $\Sigma ↦{\int }_{\Omega }\mathrm{dist}\left(x,\Sigma \right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\nu$ (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.

DOI : https://doi.org/10.1051/cocv:2008034
Classification : 90B80,  90B85,  49J45,  46N10,  60K30
Mots clés : location problem, facility location, Fermat-Weber problem, $k$-median problem, sequential allocation, average distance functional, optimal transportation
@article{COCV_2009__15_3_509_0,
author = {Brancolini, Alessio and Buttazzo, Giuseppe and Santambrogio, Filippo and Stepanov, Eugene},
title = {Long-term planning versus short-term planning in the asymptotical location problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {509--524},
publisher = {EDP-Sciences},
volume = {15},
number = {3},
year = {2009},
doi = {10.1051/cocv:2008034},
zbl = {1169.90386},
mrnumber = {2542570},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_3_509_0/}
}
Brancolini, Alessio; Buttazzo, Giuseppe; Santambrogio, Filippo; Stepanov, Eugene. Long-term planning versus short-term planning in the asymptotical location problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 509-524. doi : 10.1051/cocv:2008034. http://www.numdam.org/item/COCV_2009__15_3_509_0/

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