Augmented Lagrangian Methods for Transport Optimization, Mean-field Games and Degenerate Pdes

Many problems from mass transport can be reformulated as variational problems under a prescribed divergence constraint (static problems) or subject to a time dependent continuity equation which again is a divergence constraint but in time and space. A large class of Mean-Field Games introduced by Lasry and Lions may also be interpreted as a generalisation of the time-dependent optimal transport problem. Following Benamou and Brenier [BB00], we show that augmented Lagrangian methods are well-suited to treat such convex but neither smooth nor strictly convex problems. It includes in particular Monge’s original optimal transport problem. A finite element discretization and implementation of the method is used to provide numerical simulations and a convergence study.