# Complexity of Proximal augmented Lagrangian for nonconvex optimization with nonlinear equality constraints

We analyze worst-case complexity of a Proximal augmented Lagrangian (Proximal AL) framework for nonconvex optimization with nonlinear equality constraints. When an approximate first-order (second-order) optimal point is obtained in the subproblem, an $\epsilon$ first-order (second-order) optimal point for the original problem can be guaranteed within $\mathcal{O}(1/ \epsilon^{2 - \eta})$ outer iterations (where $\eta$ is a user-defined parameter with $\eta\in[0,2]$ for the first-order result and $\eta \in [1,2]$ for the second-order result) when the proximal term coefficient $\beta$ and penalty parameter $\rho$ satisfy $\beta = \mathcal{O}(\epsilon^\eta)$ and $\rho = \Omega (1/\epsilon^\eta)$, respectively. We also investigate the total iteration complexity and operation complexity when a Newton-conjugate-gradient algorithm is used to solve the subproblems. Finally, we discuss an adaptive scheme for determining a value of the parameter $\rho$ that satisfies the requirements of the analysis.