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| Main Authors: | , , |
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| Format: | Preprint |
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2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2011.13263 |
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| _version_ | 1866929739225104384 |
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| author | Artstein-Avidan, Shiri Sadovsky, Shay Wyczesany, Katarzyna |
| author_facet | Artstein-Avidan, Shiri Sadovsky, Shay Wyczesany, Katarzyna |
| contents | In this note, we present a unified approach to the problem of existence of a potential for the optimal transport problem with respect to non-traditional cost functions, that is, costs that assume infinite values. We establish a new method that relies on proving solvability of a special (possibly infinite) family of linear inequalities. When the index set of this family is countable, we give a necessary and sufficient condition on the coefficients that assures the existence of a solution, and which, in the setting of transport theory, we call $c$-path-boundedness. In the case of an uncountable index set, one needs an additional assumption for solvability. We propose a sufficient condition in this case. We note that any set admitting a potential must be $c$-path-bounded, and this condition replaces $c$-cyclic monotonicity from the classical theory, i.e. when the cost is real-valued. Our method also gives a new and elementary proof for the classical results of Rockafellar, Rochet and Rüschendorf. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_13263 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A Rockafellar-type theorem for non-traditional costs Artstein-Avidan, Shiri Sadovsky, Shay Wyczesany, Katarzyna Metric Geometry Functional Analysis 49K27, 47A63 In this note, we present a unified approach to the problem of existence of a potential for the optimal transport problem with respect to non-traditional cost functions, that is, costs that assume infinite values. We establish a new method that relies on proving solvability of a special (possibly infinite) family of linear inequalities. When the index set of this family is countable, we give a necessary and sufficient condition on the coefficients that assures the existence of a solution, and which, in the setting of transport theory, we call $c$-path-boundedness. In the case of an uncountable index set, one needs an additional assumption for solvability. We propose a sufficient condition in this case. We note that any set admitting a potential must be $c$-path-bounded, and this condition replaces $c$-cyclic monotonicity from the classical theory, i.e. when the cost is real-valued. Our method also gives a new and elementary proof for the classical results of Rockafellar, Rochet and Rüschendorf. |
| title | A Rockafellar-type theorem for non-traditional costs |
| topic | Metric Geometry Functional Analysis 49K27, 47A63 |
| url | https://arxiv.org/abs/2011.13263 |