Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints

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, both assumptions. ALM may experience oscillations and divergence when underlying problem simultaneously nonconvex nonsmooth. In this paper, we consider linearly with a (in particular, weakly convex) nonsmooth objective. We modify to use Moreau envelope establish its conditions that are weaker than those in literature. call it (MEAL) method. also show iteration complexity MEAL $$o(\varepsilon ^{-2})$$ yield an $$\varepsilon $$ -accurate first-order stationary point. whole sequence (regardless initial guess) rate Kurdyka–?ojasiewicz property assumed. Moreover, subproblem no closed-form solution difficult solve, propose two practical variants MEAL, inexact version called iMEAL approximate proximal update, linearized LiMEAL composite Their established.