Stability for Constrained Minimax Optimization (CSIAM Transactions on Applied Mathematics, 2023)
Minimax optimization problems are an important class of optimization problems arising from both modern machine learning and from traditional research areas.We focus on the stability of constrained minimax optimization problems based on thenotion of local minimax point by Dai and Zhang (2020). Firstly, we extend the classicalJacobian uniqueness conditions of nonlinear programming to the constrained minimax problem and prove that this set of properties is stable with respect to small $\mathcal{C}^2$ perturbation. Secondly, we provide a set of conditions, called Property A, which doesnot require the strict complementarity condition for the upper level constraints. Finally, we prove that Property A is a sufficient condition for the strong regularity of theKurash-Kuhn-Tucker (KKT) system at the KKT point, and it is also a sufficient condition for the local Lipschitzian homeomorphism of the Kojima mapping near the KKTpoint.
Keywords: Constrained minimax optimization; Jacobian uniqueness conditions; strong regularity; strong sufficient optimality condition; Kojima mapping; local Lipschitzian homeomorphism.