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, $$\epsilon $$ original problem can be guaranteed within $${\mathcal {O}}(1/ \epsilon ^{2 - \eta })$$ outer iterations (where $$\eta user-defined parameter \in [0,2]$$ result and [1,2]$$ second-order result) when proximal term coefficient $$\beta penalty $$\rho satisfy = {\mathcal {O}}(\epsilon ^\eta )$$ \varOmega (1/\epsilon , respectively. also investigate total iteration operation Newton-conjugate-gradient algorithm used to solve subproblems. Finally, we discuss adaptive scheme determining value that satisfies requirements analysis.