A new class of costs for optimal transport planning

By J. Alibert, Guy Bouchitté, Thierry Champion. European journal of applied mathematics, 2018

50 citations recorded by the source corpus.

We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ($G(x,p)=\int c(x,y)dp$) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/f8b9e015c8ae3f22912ee38a1a043f357e92fe5f

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In Applied Mathematics

Citations in the corpus

Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems (2017, unknown): “)j%(dx) under the constraint that Ris a Markov kernel such that R= : Note that this problem is a particular case of the general transport costs considered by Gozlan et al. [15] and Alibert et al. [3], who are interested in duality results and by Backho- Veraguas et al. [5] who deal with existence o”
The Wasserstein space of stochastic processes (2021, background): “the survey of Leonard [55]), martingale optimal transport, the weak transport problem (see [36, 6] among many others), the theory of linear transfers ([25]), and mathematical finance / martingale optimal transport, we refer to [10, 15, 2, 12].”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, background): “[2] in the case where X , Y are compact spaces.”
Applications of weak transport theory (2020, unknown): “Overview The optimal transport problem for weak costs was first introduced by Gozlan, Roberto, Samson and Tetali [34] and has immediately generated interest in several groups of researchers, see [45, 46, 29, 32, 1, 2] among others.”
Stability of martingale optimal transport and weak optimal transport (2019, unknown): “Marton [34, 33] and Talagrand [36, 37]), weak transport problems appear also in a number of further topics, including martingale transport [2, 4, 16, 8, 9], the causal transport problem [5, 1], and stability in mathematical finance [6].”
On a mixture of Brenier and Strassen Theorems (2018, background): “ recovered and completed by Alfonsi, Corbetta and Jourdain in [1], where an explicit expression is given for ¯µ(see [1, Proposition 3.4]). 16 NATHAEL GOZLAN AND NICOLAS JUILLET (2)In the recent paper [2] (see in particular the section 5.2 of [2]), Alibert, Bouchitt´e and Champion consider the family”