Duality for the $L^{\infty }$ optimal transport problem
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- by E. N. Barron, M. Bocea and R. R. Jensen PDF
- Trans. Amer. Math. Soc. 369 (2017), 3289-3323 Request permission
Abstract:
We derive the dual of the relaxed Monge-Kantorovich optimal mass transport problem in $L^{\infty }$ in which one seeks to minimize $\mu$-$\mathrm {ess sup}_{(x,y)\in \mathbb {R}^N \times \mathbb {R}^N} c(x,y)$ over Borel probability measures $\mu$ with given marginals $P_0, P_1.$ Several formulations of the dual problem are obtained using various techniques including quasiconvex duality. We also consider weighted optimal transport in $L^{\infty }$ and we identify the form of the dual in the Lagrangian cost setting for both integral and essential supremum costs along trajectories. Finally, we prove a duality formula that relates a maximization problem which arises naturally in the $L^{\infty }$ calculus of variations with a family of optimal partial transport problems.References
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Additional Information
- E. N. Barron
- Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, Illinois 60660
- MR Author ID: 31685
- Email: ebarron@luc.edu
- M. Bocea
- Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, Illinois 60660
- MR Author ID: 617221
- Email: mbocea@luc.edu
- R. R. Jensen
- Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, Illinois 60660
- MR Author ID: 205502
- Email: rjensen@luc.edu
- Received by editor(s): October 1, 2014
- Received by editor(s) in revised form: April 30, 2015
- Published electronically: September 15, 2016
- Additional Notes: The research of the first and third authors was partially supported by the National Science Foundation under Grants No. DMS-1008602 and DMS-1515871
The second author’s research was partially supported by the National Science Foundation under Grants No. DMS-1156393 and DMS-1515871 - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3289-3323
- MSC (2010): Primary 35F21, 49J35, 49J45, 49K30, 49L25, 49Q20
- DOI: https://doi.org/10.1090/tran/6759
- MathSciNet review: 3605972