Semi-infinite Nonconvex Constrained Min-Max Optimization

Cody Melcher, Zeinab Alizadeh, Lindsey Hiett, Afrooz Jalilzadeh, Erfan Yazdandoost Hamedani

Keywords: Semi-infinite Programming, Nonconvex Constrained Optimization, Min-Max Optimization

TL;DR: Developing and analyzing a novel inexact dynamic barrier primal-dual algorithm for solving nonconvex min-max optimization problems with infinite constraints.

Abstract: 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 $\mathcal{O}(\epsilon^{-3})$, $\mathcal{O}(\epsilon^{-6\theta})$, and $\mathcal{O}(\epsilon^{-3\theta/(1-\theta)})$ iteration complexities to achieve an $\epsilon$-approximate stationarity, infeasibility, and complementarity slackness, respectively. Numerical experiments on robust multitask learning with task priority further illustrate the practical effectiveness of the algorithm.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 23047