# The proximal augmented Lagrangian method for nonsmooth composite optimization

arXiv: https://arxiv.org/abs/1610.04514 · HTML: https://mlsanity.com/paper/1610.04514/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2016.

## Built on by (most cited first, 11 of 11)

- [Faster Projection-Free Augmented Lagrangian Methods via Weak Proximal Oracle](https://mlsanity.com/paper/2210.13968/) (2022)
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- [Fixed-Time Convergence for a Class of Nonconvex-Nonconcave Min-Max Problems](https://mlsanity.com/paper/2207.12845/) (2022)
- [Private Networked Federated Learning for Nonsmooth Objectives](https://mlsanity.com/paper/2306.14012/) (2023)
- [Distributed Optimization via Kernelized Multi-armed Bandits](https://mlsanity.com/paper/2312.04719/) (2023)
- [Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems](https://mlsanity.com/paper/2408.15969/) (2024)
- [From exponential to finite/fixed-time stability: Applications to optimization](https://mlsanity.com/paper/2409.11713/) (2024)
- [Global Optimality for Constrained Exploration via Penalty Regularization](https://mlsanity.com/paper/2604.28144/) (2026)

## Abstract (arXiv)

We study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term. We introduce an auxiliary variable to separate the objective function components and utilize the Moreau envelope of the regularization term to derive the proximal augmented Lagrangian - a continuously differentiable function obtained by constraining the augmented Lagrangian to the manifold that corresponds to the explicit minimization over the variable in the nonsmooth term. The continuous differentiability of this function with respect to both primal and dual variables allows us to leverage the method of multipliers (MM) to compute optimal primal-dual pairs by solving a sequence of differentiable problems. The MM algorithm is applicable to a broader class of problems than proximal gradient methods and it has stronger convergence guarantees and a more refined step-size update rules than the alternating direction method of multipliers. These features make it an attractive option for solving structured optimal control problems. We also develop an algorithm based on the primal-descent dual-ascent gradient method and prove global (exponential) asymptotic stability when the differentiable component of the objective function is (strongly) convex and the regularization term is convex. Finally, we identify classes of problems for which the primal-dual gradient flow dynamics are convenient for distributed implementation and compare/contrast our framework to the existing approaches.