  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.
