An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints

TL;DR

This paper introduces a practical inexact augmented Lagrangian method for nonconvex problems with nonlinear constraints, providing complexity analysis and numerical validation on large-scale machine learning tasks.

Contribution

It develops a new inexact augmented Lagrangian framework with complexity guarantees for finding stationary points in nonconvex constrained optimization.

Findings

Achieves $ ilde{O}(1/ ext{epsilon}^4)$ complexity for first-order stationary points.

Achieves $ ilde{O}(1/ ext{epsilon}^5)$ complexity for second-order stationary points.

Demonstrates effectiveness on large-scale machine learning problems and verifies geometric conditions.

Abstract

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.