Duality for the 𝐿^{∞} optimal transport problem

Barron, E.; Bocea, M.; Jensen, R.

doi:10.1090/tran/6759

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.

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