  The augmented Lagrangian method (ALM) is one of the most useful methods for
constrained optimization. Its convergence has been well established under
convexity assumptions or smoothness assumptions, or under both assumptions. ALM
may experience oscillations and divergence facing nonconvexity and
nonsmoothness simultaneously. In this paper, we consider the linearly
constrained problem with a nonconvex (in particular, weakly convex) and
nonsmooth objective. We modify ALM using a Moreau envelope of the augmented
Lagrangian and establish its convergence under conditions that are weaker than
those in the literature. We call it the Moreau envelope augmented Lagrangian
(MEAL) method. We also show that the iteration complexity of MEAL is
$o(\varepsilon^{-2})$ to yield an $\varepsilon$-accurate first-order stationary
point. We establish its whole sequence convergence (regardless of the initial
guess) and a rate when a Kurdyka-{\L}ojasiewicz property is assumed. Moreover,
when the subproblem of MEAL has no closed-form solution and is difficult to
solve, we propose two practical variants of MEAL, an inexact version called
iMEAL with an approximate proximal update, and a linearized version called
LiMEAL for the constrained problem with a composite objective. Their
convergence is also established.
