Variational models of robust optimal transport

TL;DR

This work addresses robust optimal transport by introducing two variational formulations that incorporate random damages and recovery plans. The Eulerian formulation uses an unoriented 1-rectifiable network $\mu$ with recovery currents $\{T_i\}$, while the Lagrangian formulation uses oriented traffic plans $P$ with subplans $P_i$ and a damage payoff. The authors prove the existence of minimizers in both settings and compare the models via examples, illustrating how orientation and the admissible class of damages influence well-posedness. Overall, the paper extends branched transport theory to redundancy-aware networks and provides a variational framework for designing robust transport infrastructures.

Abstract

This paper introduces two variational formulations for a model of robust optimal transport, that is, the problem of designing optimal transport networks that are resilient to potential damages, balancing construction costs against the benefit of maintaining partial functionality when parts of the network are damaged. We propose a Eulerian formulation, where the network is modeled by a rectifiable measure and recovery plans are represented by 1-dimensional normal currents. This framework allows for changes in the direction of the transportation in response to damages but restricts damages to be characteristic functions of closed sets. We also propose a Lagrangian formulation, where the network is a traffic plan (that is, a measure on the space of Lipschitz curves) and recovery plans are sub-traffic plans. This approach prescribes the network's orientation but allows for a wider class of damages. We prove existence of minimizers in both settings. The two models are compared through examples that illustrate their main differences: the Eulerian formulation's necessity for an unoriented network to achieve existence, the Lagrangian formulation's ability to handle general damages and its requirement for a positive distance between the supports of the source and target measures.

Figures (4)

• Figure 1: The measures $\nu^-$ (in red) and $\nu^+$ (in blue) and the sets $S_1,S_2$ of the example \ref{['es:non_existence']}. The pattern $\hat{T}_1$, $\hat{T}_2$ are individually optimal for the pay-off term, but they cannot be both contained in any oriented network $T$, since the two segments between $(-1,-1)$ and $(1,-1)$ have opposite orientations.
• Figure 2: The situation described in Example \ref{['ex:non_continuous']}: the shape of the damage $f$ forces the optimal recovery plan to contain two overlapping curves with different orientations.
• Figure 3: A representation of Example \ref{['ex:distance']}: each damage $f_i$ forces to choose a recovery plan supported on $\gamma_i$, while the pay-off forces any minimizing traffic plan $P$ to use all of them, hence obtaining infinite mass.
• Figure 4: Example \ref{['ex:limit']}: if $\phi$ is bounded and the curves $\gamma_i$ transporting $\nu^-$ to $\nu^+$ are "almost overlapping", then for a minimizer it would be convenient to charge all of them, obtaining a traffic plan with infinite mass.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1: good decomposition
• Example 5.1
• Example 5.2
• Example 5.3
• Example 5.4