Mohit Bansil, Jun Kitagawa
Abstract:
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.
Submitted June 22, 2022. Published March 2, 2023.
Math Subject Classifications: 49J45, 49K40, 49Q22.
Key Words: Optimal transport; gamma convergence; storage fees; queue penalization.
DOI: https://doi.org/10.58997/ejde.2023.22
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Mohit Bansil Department of Mathematics University of California Los Angeles, CA 90095, USA email: mbansil@math.ucla.edu |
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Jun Kitagawa Department of Mathematics Michigan State University 619 Red Cedar Road, C212 Wells Hall East Lansing, MI 48824, USA email: kitagawa@math.msu.edu |
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