Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

9 Scopus Citations
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Author(s)

  • Jinshan Zeng
  • Wotao Yin
  • Ding-Xuan Zhou

Detail(s)

Original languageEnglish
Article number61
Journal / PublicationJournal of Scientific Computing
Volume91
Issue number2
Online published7 Apr 2022
Publication statusPublished - May 2022

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(ε- 2) to yield an ε-accurate first-order stationary point. We establish its whole sequence convergence (regardless of the initial guess) and a rate when a Kurdyka–Łojasiewicz 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.

Research Area(s)

  • Augmented Lagrangian method, Kurdyka–Łojasiewicz inequality, Moreau envelope, Nonconvex nonsmooth optimization, Proximal augmented Lagrangian method

Citation Format(s)

Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints. / Zeng, Jinshan; Yin, Wotao; Zhou, Ding-Xuan.
In: Journal of Scientific Computing, Vol. 91, No. 2, 61, 05.2022.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review