# Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints

arXiv: https://arxiv.org/abs/2101.08519 · HTML: https://mlsanity.com/paper/2101.08519/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jan 2021.

## Built on by (most cited first, 10 of 10)

- [A Single-Loop Gradient Descent and Perturbed Ascent Algorithm for Nonconvex Functional Constrained Optimization](https://mlsanity.com/paper/2207.05650/) (2022)
- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [Smoothing ADMM for Sparse-Penalized Quantile Regression with Non-Convex Penalties](https://mlsanity.com/paper/2309.03094/) (2023)
- [A Riemannian ADMM](https://mlsanity.com/paper/2211.02163/) (2022)
- [Damped Proximal Augmented Lagrangian Method for weakly-Convex Problems with Convex Constraints](https://mlsanity.com/paper/2311.09065/) (2023)
- [Analysis of regularized federated learning](https://mlsanity.com/paper/2411.01548/) (2024)
- [Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions](https://mlsanity.com/paper/2502.19764/) (2025)
- [Moreau Envelope ADMM for Decentralized Weakly Convex Optimization](https://mlsanity.com/paper/2308.16752/) (2023)
- [A Penalty-Based Guardrail Algorithm for Non-Decreasing Optimization with Inequality Constraints](https://mlsanity.com/paper/2405.01984/) (2024)
- [ADMM for Structured Fractional Minimization](https://mlsanity.com/paper/2411.07496/) (2024)

## Abstract (arXiv)

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.