An optimal transport problem with storage fees
DOI:
https://doi.org/10.58997/ejde.2023.22Abstract
We establish basic properties of a variant of the semi-discrete optimal transport problem in a relatively general setting. In this problem, one is given an absolutely continuous source measure and cost function, along with a finite set which will be the support of the target measure, and a “storage fee” function. The goal is to find a map for which the total transport cost plus the storage fee evaluated on the masses of the pushforward of the source measure is minimized. We prove existence and uniqueness for the problem, derive a dual problem for which strong duality holds, and give a characterization of dual maximizers and primal minimizers. Additionally, we find some stability results for minimizers and a Γ-convergence result as the target set becomes denser and denser in a continuum domain.
For more information see https://ejde.math.txstate.edu/Volumes/2023/22/abstr.html
References
Mohit Bansil, Jun Kitagawa; A Newton Algorithm for Semidiscrete Optimal Transport with Storage Fees, SIAM J. Optim. 31 (2021), no. 4, 2586-2613. MR 4329988 https://doi.org/10.1137/20M1357226
Patrick Billingsley; Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication. MR 1700749 https://doi.org/10.1002/9780470316962
V. I. Bogachev; Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655 https://doi.org/10.1007/978-3-540-34514-5
Andrea Braides; Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications, vol. 22, Oxford University Press, Oxford, 2002. MR 1968440
Guillaume Carlier, Lina Mallozzi; Optimal monopoly pricing with congestion and random utility via partial mass transport, J. Math. Anal. Appl. 457 (2018), no. 2, 1218-1231. MR 3705351 https://doi.org/10.1016/j.jmaa.2017.01.003
Gianluca Crippa, Chloé Jimenez, Aldo Pratelli; Optimum and equilibrium in a transport problem with queue penalization effect, Adv. Calc. Var. 2 (2009), no. 3, 207-246. MR 2537021 https://doi.org/10.1515/ACV.2009.009
Claude Dellacherie, Paul-André Meyer; Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 521810
Lina Mallozzi, Antonia Passarelli di Napoli; Optimal transport and a bilevel location- allocation problem, J. Global Optim. 67 (2017), no. 1-2, 207-221. MR 3596835 https://doi.org/10.1007/s10898-015-0347-7
R. Tyrrell Rockafellar; Convex analysis, Princeton University Press, 1970. https://doi.org/10.1515/9781400873173
Filippo Santambrogio; Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications, vol. 87, Birkhäuser/Springer, Cham, 2015, Calculus of variations, PDEs, and modeling. MR 3409718 https://doi.org/10.1007/978-3-319-20828-2
Cédric Villani; Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483 (2004e:90003)
Cédric Villani; Optimal transport: Old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. MR 2459454 (2010f:49001) https://doi.org/10.1007/978-3-540-71050-9
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