Some computational aspects on (martingale) optimal transportation

The optimal transport is concerned with transferring mass from one location to another in such a way as to optimize a given criterion. The talk contains two parts. Motivated by the robust finance, a constrained optimal transport problem has recently been taken into consideration, where an additional martingale constraint is imposed on the transport's dynamics. We develop two numerical methods to solve such problems: using discretization and linear programming applied to the primal side and using penalization and deep neural networks applied to the dual side. In the second part, we consider a parametrized semi-discrete optimal transport problem. The optimization problem under consideration has a common formulation with various machine learning problems, while the objective function's regularity is not clear. We adopt Lloyd's algorithm to this optimization problem and obtain some convergence results.