An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints

摘要: We propose a practical inexact augmented Lagrangian method (iALM) for nonconvex problems with nonlinear constraints. characterize the total computational complexity of our subject to verifiable geometric condition, which is closely related Polyak-Lojasiewicz and Mangasarian-Fromovitz conditions. In particular, when first-order solver used inner iterates, we prove that iALM finds stationary point $\tilde{\mathcal{O}}(1/\epsilon^3)$ calls oracle. If, in addition, problem smooth second-order $\tilde{\mathcal{O}}(1/\epsilon^5)$ These results match known theoretical literature. We also provide strong numerical evidence on large-scale machine learning problems, including Burer-Monteiro factorization semidefinite programs, novel relaxation standard basis pursuit template. For these examples, show how verify condition.