  Minimax optimization problems arises from both modern machine learning
including generative adversarial networks, adversarial training and multi-agent
reinforcement learning, as well as from tradition research areas such as saddle
point problems, numerical partial differential equations and optimality
conditions of equality constrained optimization. For the unconstrained
continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019)
carefully considered the very basic question: what is a proper definition of
local optima of a minimax optimization problem, and proposed a proper
definition of local optimality called local minimax. We shall extend the
definition of local minimax point to constrained nonconvex-nonconcave minimax
optimization problems. By analyzing Jacobian uniqueness conditions for the
lower-level maximization problem and the strong regularity of
Karush-Kuhn-Tucker conditions of the maximization problem, we provide both
necessary optimality conditions and sufficient optimality conditions for the
local minimax points of constrained minimax optimization problems.
