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

Brancolini, Alessio; Buttazzo, Giuseppe; Santambrogio, Filippo; Stepanov, Eugene

doi:10.1051/cocv:2008034

Brancolini, Alessio; Buttazzo, Giuseppe; Santambrogio, Filippo; Stepanov, Eugene

ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 3, p. 509-524

Abstract

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

MR 2542570 | Zbl 1169.90386 | 2 citations in Numdam

DOI : https://doi.org/10.1051/cocv:2008034
Classification: 90B80, 90B85, 49J45, 46N10, 60K30
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     pages = {509-524},
     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, Volume 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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