# On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization

## TL;DR

This paper analyzes the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex constrained optimization, providing bounds on iterations and evaluations depending on penalty parameter behavior and problem constraints.

## Contribution

It establishes new complexity bounds for the Augmented Lagrangian method in nonconvex settings, including cases with bounded and unbounded penalty parameters, and different types of constraints.

## Key findings

- Bounded penalty parameters yield $ ext{O}(| ext{log}( ext{epsilon})|)$ iteration complexity.
- Unbounded penalty parameters lead to $ ext{O}( ext{epsilon}^{-2/( ext{alpha}-1)})$ iteration complexity.
- Specific evaluation complexity bounds are derived for linearly constrained problems with different inner solver orders.

## Abstract

In this paper we study the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded, we prove a complexity bound of $\mathcal{O}(|\log(\epsilon)|)$ outer iterations for the referred algorithm to generate an $\epsilon$-approximate KKT point, for $\epsilon\in (0,1)$. When the penalty parameters are unbounded, we prove an outer iteration complexity bound of $\mathcal{O}\left(\epsilon^{-2/(\alpha-1)}\right)$, where $\alpha>1$ controls the rate of increase of the penalty parameters. For linearly constrained problems, these bounds yield to evaluation complexity bounds of $\mathcal{O}(|\log(\epsilon)|^{2}\epsilon^{-2})$ and $\mathcal{O}\left(\epsilon^{-\left(\frac{2(2+\alpha)}{\alpha-1}+2\right)}\right)$, respectively, when appropriate first-order methods ($p=1$) are used to approximately solve the unconstrained subproblems at each iteration. In the case of problems having only linear equality constraints, the latter bounds are improved to $\mathcal{O}(|\log(\epsilon)|^{2}\epsilon^{-(p+1)/p})$ and $\mathcal{O}\left(\epsilon^{-\left(\frac{4}{\alpha-1}+\frac{p+1}{p}\right)}\right)$, respectively, when appropriate $p$-order methods ($p\geq 2$) are used as inner solvers.