Optimality Conditions for Constrained Minimax Optimization (CSIAM Transactions on Applied Mathematics, 2020)
Minimax optimization problems arises from both modern machine learningincluding 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 equalityconstrained optimization. For the unconstrained continuous nonconvex-nonconcavesituation, 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 shallextend the definition of local minimax point to constrained nonconvex-nonconcaveminimax optimization problems. By analyzing Jacobian uniqueness conditions for thelower-level maximization problem and the strong regularity of Karush-Kuhn-Tuckerconditions of the maximization problem, we provide both necessary optimality conditions and sufficient optimality conditions for the local minimax points of constrainedminimax optimization problems.