This talk considers nonsmooth nonconvex-nonconcave min-max optimization problems with convex feasible sets. We discuss the existence of local saddle points, global minimax points and local minimax points, and study the optimality conditions for local minimax points. We show the existence of local saddle points and global minimax points of the convex-concave saddle point problem with cardinality penalties and the relations with its continuous relaxation problems. Moreover, we give an explicit formula for the value function of the inner maximization problem of a class of robust nonlinear least square problems and complexity bound for finding an approximate first order stationary point. A smoothing quasi-Newton subspace trust region algorithm is presented for training generative adversarial networks as nonsmooth nonconvex-nonconcave min-max optimization problems. Examples of retinal vessel segmentation in fundoscopic images are used to illustrate the efficiency of the algorithm.