Given the probability measure over the given region , we consider the optimal location of a set composed by points in in order to minimize the average distance (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all 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 , although the optimization costs in both cases have the same asymptotic orders of vanishing.
Classification : 90B80, 90B85, 49J45, 46N10, 60K30
Mots clés : location problem, facility location, Fermat-Weber problem, -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/
[1] Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | MR 1387558 | Zbl 0957.49029
,[2] Gradient flows in metric spaces and in the spaces of probability measures, Lectures in Mathematics. ETH Zurich, Birkhäuser (2005). | MR 2129498 | Zbl 1090.35002
, and ,[3] Asymptotique d'un problème de positionnement optimal. C. R. Acad. Sci. Paris Ser. I 335 (2002) 1-6. | MR 1947712 | Zbl 1041.90025
, and ,[4] Optimal networks for mass transportation problems. ESAIM: COCV 11 (2005) 88-101. | Numdam | MR 2110615 | Zbl 1103.49002
and ,[5] Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem. Ann. Scuola Norm. Sup. Cl. Sci. II (2003) 631-678. | Numdam | MR 2040639 | Zbl 1127.49031
and ,[6] Minimization problems for average distance functionals, in Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, D. Pallara Ed., Quaderni di Matematica 14, Seconda Universita di Napoli (2004) 47-83. | MR 2118415 | Zbl 1089.49040
and ,[7] Optimal transportation problems with free Dirichlet regions, Progress in Nonlinear Differential Equations and their Applications 51. Birkhäuser (2002) 41-65. | MR 2197837 | Zbl 1055.49029
, and ,[8] An introduction to -convergence. Birkhauser, Basel (1992). | MR 1201152 | Zbl 0816.49001
,[9] Lagerungen in der Ebene auf der Kugel und im Raum, Die Grundlehren der Math. Wiss. 65. Springer-Verlag, Berlin (1953). | MR 57566 | Zbl 0052.18401
,[10] Hexagonal economic regions solve the location problem. Amer. Math. Monthly 109 (2002) 165-172. | MR 1903153 | Zbl 1026.90059
and ,[11] -convergence for the irrigation problem. J. Convex Anal. 12 (2005) 145-158. | MR 2135803 | Zbl 1076.49024
and ,[12] Qualitative properties of maximum distance minimizers and average distance minimizers in . J. Math. Sciences (N.Y.) 122 (2004) 105-122. | MR 2084183 | Zbl 1099.49029
and ,[13] Blow-up of optimal sets in the irrigation probem. J. Geom. Anal. 15 (2005) 343-362. | MR 2152486 | Zbl 1115.49029
and ,[14] Partial geometric regularity of some optimal connected transportation networks. J. Math. Sciences (N.Y.) 132 (2006) 522-552. | MR 2197342 | Zbl 1121.49039
,[15] The -center location. Location Sci. 4 (1996) 69-82. | Zbl 0917.90233
and ,[16] Using Voronoi diagrams, in Facility location: a survey of applications and methods, Z. Drezner Ed., Springer Series in Operations Research, Springer Verlag (1995) 103-118.
and ,[17] Sequential location-allocation of public facilities in one- and two-dimensional space: comparison of several policies. Math. Program. Ser. B 52 (1991) 125-146. | MR 1111083 | Zbl 0734.90051
, and ,[18] http://cvgmt.sns.it/papers/brabutsan06/.