  We study MinMax solution methods for a general class of optimization problems
related to (and including) optimal transport. Theoretically, the focus is on
fitting a large class of problems into a single MinMax framework and
generalizing regularization techniques known from classical optimal transport.
We show that regularization techniques justify the utilization of neural
networks to solve such problems by proving approximation theorems and
illustrating fundamental issues if no regularization is used. We further study
the relation to the literature on generative adversarial nets, and analyze
which algorithmic techniques used therein are particularly suitable to the
class of problems studied in this paper. Several numerical experiments showcase
the generality of the setting and highlight which theoretical insights are most
beneficial in practice.
