  This paper considers a class of nonsmooth nonconvex-nonconcave min-max
problems in machine learning and games. We first provide sufficient conditions
for the existence of global minimax points and local minimax points. Next, we
establish the first-order and second-order optimality conditions for local
minimax points by using directional derivatives. These conditions reduce to
smooth min-max problems with Fr{\'e}chet derivatives. We apply our theoretical
results to generative adversarial networks (GANs) in which two neural networks
contest with each other in a game. Examples are used to illustrate applications
of the new theory for training GANs.
