  Convergence to a saddle point for convex-concave functions has been studied
for decades, while recent years has seen a surge of interest in non-convex
(zero-sum) smooth games, motivated by their recent wide applications. It
remains an intriguing research challenge how local optimal points are defined
and which algorithm can converge to such points. An interesting concept is
known as the local minimax point, which strongly correlates with the
widely-known gradient descent ascent algorithm. This paper aims to provide a
comprehensive analysis of local minimax points, such as their relation with
other solution concepts and their optimality conditions. We find that local
saddle points can be regarded as a special type of local minimax points, called
uniformly local minimax points, under mild continuity assumptions. In
(non-convex) quadratic games, we show that local minimax points are (in some
sense) equivalent to global minimax points. Finally, we study the stability of
gradient algorithms near local minimax points. Although gradient algorithms can
converge to local/global minimax points in the non-degenerate case, they would
often fail in general cases. This implies the necessity of either novel
algorithms or concepts beyond saddle points and minimax points in non-convex
smooth games.
