An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints

• Published in: arXiv.org
• Main Authors: Sahin, Mehmet Fatih, Eftekhari, Armin, Alacaoglu, Ahmet, Latorre, Fabian, Cevher, Volkan
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 21.04.2022
• Subjects: Complexity; Machine learning; Nonlinear programming; Optimization
• ISSN: 2331-8422

We propose a practical inexact augmented Lagrangian method (iALM) for nonconvex problems with nonlinear constraints. We characterize the total computational complexity of our method subject to a verifiable geometric condition, which is closely related to the Polyak-Lojasiewicz and Mangasarian-Fromowitz conditions. In particular, when a first-order solver is used for the inner iterates, we prove that iALM finds a first-order stationary point with \(\tilde{\mathcal{O}}(1/\epsilon^4)\) calls to the first-order oracle. If, in addition, the problem is smooth and a second-order solver is used for the inner iterates, iALM finds a second-order stationary point with \(\tilde{\mathcal{O}}(1/\epsilon^5)\) calls to the second-order oracle, which matches the known theoretical complexity result in the literature. We also provide strong numerical evidence on large-scale machine learning problems, including the Burer-Monteiro factorization of semidefinite programs, and a novel nonconvex relaxation of the standard basis pursuit template. For these examples, we also show how to verify our geometric condition.