Optimal partial transport problem with Lagrangian costs

Igbida, Noureddine; Nguyen, Van Thanh

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 2109-2132.

We introduce a dual dynamical formulation for the optimal partial transport problem with Lagrangian costs

$$c_L(x,y):=\underset{\begin{array}{c}\xi \in \mathrm{Lip}([0,1];{ℝ}^N)\end{array}}{inf}\{{\int }_0^{1}L(\xi \left(t\right),\dot{\xi }\left(t\right))\mathrm{d}t:\xi \left(0\right)=x,\xi \left(1\right)=y\}$$

based on a constrained Hamilton–Jacobi equation. Optimality condition is given that takes the form of a system of PDEs in some way similar to constrained mean field games. The equivalent formulations are then used to give numerical approximations to the optimal partial transport problem via augmented Lagrangian methods. One of advantages is that the approach requires only values of $L$ and does not need to evaluate $c_L(x,y)$, for each pair of endpoints $x$ and $y$, which comes from a variational problem. This method also provides at the same time active submeasures and the associated optimal transportation.

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