  Nonconvex-nonconcave minimax problems have found numerous applications in
various fields including machine learning. However, questions remain about what
is a good surrogate for local minimax optimum and how to characterize the
minimax optimality. Recently Jin, Netrapalli, and Jordan (ICML 2020) introduced
a concept of local minimax point and derived optimality conditions for the
smooth and unconstrained case. In this paper, we introduce the concept of calm
local minimax point, which is a local minimax point with a calm radius
function. With the extra calmness property we obtain first and second-order
sufficient and necessary optimality conditions for a very general class of
nonsmooth nonconvex-nonconcave minimax problem. Moreover we show that the calm
local minimax optimality and the local minimax optimality coincide under a weak
sufficient optimality condition for the maximization problem. This equivalence
allows us to derive stronger optimality conditions under weaker assumptions for
local minimax optimality.
