Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints

TL;DR

This paper introduces MEAL, a modified augmented Lagrangian method using Moreau envelopes for nonconvex, nonsmooth problems with linear constraints, proving convergence under weaker conditions and providing practical variants.

Contribution

It proposes the MEAL method, which ensures convergence for nonconvex, nonsmooth constrained problems under weaker assumptions than existing methods.

Findings

MEAL converges to first-order stationary points with iteration complexity o(ε^{-2})

Whole sequence convergence is established regardless of initial guess

Practical variants iMEAL and LiMEAL are proposed with proven convergence

Abstract

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 when the underlying problem is simultaneously nonconvex and nonsmooth. In this paper, we consider the linearly constrained problem with a nonconvex (in particular, weakly convex) and nonsmooth objective. We modify ALM to use 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-Lojasiewicz 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.