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 O(1/epsilon(2-eta)) outer iterations (where eta is a user-defined parameter with eta is an element of[0,2] for the first-order result and eta is an element of[1,2] for the second-order result) when the proximal term coefficient beta and penalty parameter rho satisfy beta=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.