The theorems by Rockafellar and Rueschendorf in optimal transport from a PDE point of view

A celebrated theorem of Rüschendorf says that a transport plan between two probability measures is optimal for the cost c if and only if its support is c-cyclically monotone. The latter condition requires nonnegativity of an alternating cost sum associated with any finite cycle in the domain of the source measure, and is somewhat mysterious.

In the talk I will focus on the quadratic cost, and show using a PDE viewpoint that the condition need only be required for a tiny subclass of cycles, associated respectively with the continuum limit and the opposite limit of two-point cycles.

This result provides a transparent explanation of Brenier's theorem (as I will of course explain), and is motivated by the fundamental open problem of computing optimal transport maps in high dimension. The latter problem can be reduced (Math. of Computation, 2024, with Maximilian Penka) to finding c-cyclical-monotonicity-violating cycles in the updating step of the Genetic Column Generation algorithm (SIAM J. on Math. of Data Science).