Stability for Constrained Minimax Optimization

TL;DR

This paper investigates the stability of constrained minimax optimization problems by extending classical conditions, introducing Property A, and establishing conditions for strong regularity and local homeomorphism of the KKT system.

Contribution

It extends Jacobian uniqueness conditions to constrained minimax problems, introduces Property A without strict complementarity, and links these to strong regularity and local homeomorphism results.

Findings

Jacobian conditions are stable under small perturbations.

Property A ensures strong regularity of the KKT system.

Property A guarantees local Lipschitzian homeomorphism of the Kojima mapping.

Abstract

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 the notion of local minimax point by Dai and Zhang (2020). Firstly, we extend the classical Jacobian uniqueness conditions of nonlinear programming to the constrained minimax problem and prove that this set of properties is stable with respect to small $C^2$ perturbation. Secondly, we provide a set of conditions, called Property A, which does not 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 the Kurash-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 KKT point.