We introduce a dual dynamical formulation for the optimal partial transport problem with Lagrangian costs
cL(x,y) := ξ∈Lip([0,1];ℝN)inf {∫01 L(ξ(t), ξ˙(t))dt : ξ(0) = x, ξ(1) = 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 cL(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.
Local Convergence of Exact and Inexact Augmented Lagrangian Methods under the Second-Order Sufficient Optimality Condition
