Perturbed Augmented Lagrangian Method Framework with Applications to Proximal and Smoothed Variants

We introduce a perturbed augmented Lagrangian method framework, which is a convenient tool for local analyses of convergence and rates of convergence of some modifications of the classical augmented Lagrangian algorithm. One example to which our development applies is the proximal augmented Lagrangian method. Previous results for this version required twice differentiability of the problem data, the linear independence constraint qualification, strict complementarity, and second-order sufficiency; or the linear independence constraint qualification and strong second-order sufficiency. We obtain a set of convergence properties under significantly weaker assumptions: once (not twice) differentiability of the problem data, uniqueness of the Lagrange multiplier, and second-order sufficiency (no linear independence constraint qualification and no strict complementarity); or even second-order sufficiency only. Another version to which the general framework applies is the smoothed augmented Lagrangian method, where the plus-function associated with penalization of inequality constraints is approximated by a family of smooth functions (so that the subproblems are twice differentiable if the problem data are). Furthermore, for all the modifications, inexact solution of subproblems is handled naturally. The presented framework also subsumes the basic augmented Lagrangian method, both exact and inexact.