On the complexity of an augmented Lagrangian method for nonconvex optimization

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 $\mathcal{O}(\epsilon ^{-2/(\alpha -1)} )$, 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 $\mathcal{O}(\epsilon ^{- (\frac{2(2+\alpha )}{\alpha -1}+2 )})$, 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 $\mathcal{O}(\epsilon ^{-(\frac{4}{\alpha -1}+\frac{p+1}{p})})$, respectively, when appropriate $p$-order methods ($p\geq 2$) are used as inner solvers.