Semi-infinite Nonconvex Constrained Min-Max Optimization
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911208797372416 |
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| author | Melcher, Cody Alizadeh, Zeinab Hiett, Lindsey Jalilzadeh, Afrooz Hamedani, Erfan Yazdandoost |
| author_facet | Melcher, Cody Alizadeh, Zeinab Hiett, Lindsey Jalilzadeh, Afrooz Hamedani, Erfan Yazdandoost |
| contents | 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 $θ\in (0,1)$, we prove that the proposed method achieves $\mathcal{O}(ε^{-3})$, $\mathcal{O}(ε^{-6θ})$, and $\mathcal{O}(ε^{-3θ/(1-θ)})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12007 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semi-infinite Nonconvex Constrained Min-Max Optimization Melcher, Cody Alizadeh, Zeinab Hiett, Lindsey Jalilzadeh, Afrooz Hamedani, Erfan Yazdandoost Optimization and Control 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 $θ\in (0,1)$, we prove that the proposed method achieves $\mathcal{O}(ε^{-3})$, $\mathcal{O}(ε^{-6θ})$, and $\mathcal{O}(ε^{-3θ/(1-θ)})$ 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. |
| title | Semi-infinite Nonconvex Constrained Min-Max Optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2510.12007 |