Complexity of Proximal Augmented Lagrangian for Nonconvex Optimization with Nonlinear Equality Constraints

Abstract

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 $$\rho = \varOmega (1/\epsilon ^\eta )$$ , 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.