Semi-infinite Nonconvex Constrained Min-Max Optimization

Semi-Infinite Programming (SIP) has emerged as a powerful framework for modeling problems with infinite constraints, however, its theoretical development in the context of nonconvex and large-scale optimization remains limited. In this paper, we investigate a class of nonconvex min-max optimization problems with nonconvex infinite constraints, motivated by applications such as adversarial robustness and safety-constrained learning.

We propose a novel inexact dynamic barrier primal-dual algorithm and establish its convergence properties. Specifically, under the assumption that the squared infeasibility residual function satisfies the Lojasiewicz inequality with exponent $\theta \in (0,1)$, we prove that the proposed method achieves $\mathcal{O}({ϵ}^{-3})$, $\mathcal{O}({ϵ}^{-6\theta })$, and $\mathcal{O}({ϵ}^{-3\theta /(1-\theta )})$ iteration complexities to achieve an $ϵ$-approximate stationarity, infeasibility, and complementarity slackness, respectively.

Numerical experiments on robust multitask learning with task priority further illustrate the practical effectiveness of the algorithm.