# A proximal augmented Lagrangian method for nonconvex optimization with equality and inequality constraints

arXiv: https://arxiv.org/abs/2509.02894 · HTML: https://mlsanity.com/paper/2509.02894/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 81 references that are in the map.
- First posted on arXiv 2 Sep 2025.

## Builds on (most cited first, 30 of 34)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Parallelizing the dual revised simplex method](https://mlsanity.com/paper/1503.01889/) (2015)
- [Clarabel: An interior-point solver for conic programs with quadratic objectives](https://mlsanity.com/paper/2405.12762/) (2024)
- [An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints](https://mlsanity.com/paper/1906.11357/) (2019)
- [Proximal Point Imitation Learning](https://mlsanity.com/paper/2209.10968/) (2022)
- [Iteration complexity of inexact augmented Lagrangian methods for constrained convex programming](https://mlsanity.com/paper/1711.05812/) (2017)
- [Rate-improved Inexact Augmented Lagrangian Method for Constrained Nonconvex Optimization](https://mlsanity.com/paper/2007.01284/) (2020)
- [General Heuristics for Nonconvex Quadratically Constrained Quadratic Programming](https://mlsanity.com/paper/1703.07870/) (2017)
- [Complexity of Proximal augmented Lagrangian for nonconvex optimization with nonlinear equality constraints](https://mlsanity.com/paper/1908.00131/) (2019)
- [A Proximal Alternating Direction Method of Multiplier for Linearly Constrained Nonconvex Minimization](https://mlsanity.com/paper/1812.10229/) (2018)
- [Structured Nonconvex and Nonsmooth Optimization: Algorithms and Iteration Complexity Analysis](https://mlsanity.com/paper/1605.02408/) (2016)
- [The proximal alternating direction method of multipliers in the nonconvex setting: convergence analysis and rates](https://mlsanity.com/paper/1801.01994/) (2018)
- [A global dual error bound and its application to the analysis of linearly constrained nonconvex optimization](https://mlsanity.com/paper/2006.16440/) (2020)
- [On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization](https://mlsanity.com/paper/1906.05622/) (2019)
- [Recurrent Neural Network Training with Convex Loss and Regularization Functions by Extended Kalman Filtering](https://mlsanity.com/paper/2111.02673/) (2021)
- [Training Recurrent Neural Networks by Sequential Least Squares and the Alternating Direction Method of Multipliers](https://mlsanity.com/paper/2112.15348/) (2021)
- [An L-BFGS-B approach for linear and nonlinear system identification under ℓ1 and group-Lasso regularization](https://mlsanity.com/paper/2403.03827/) (2024)
- [Complexity of first order inexact Lagrangian and penalty methods for conic convex programming](https://mlsanity.com/paper/1506.05320/) (2015)
- [A Simple and Efficient Algorithm for Nonlinear Model Predictive Control](https://mlsanity.com/paper/1709.06487/) (2017)
- [Iteration complexity of first-order augmented Lagrangian methods for convex conic programming](https://mlsanity.com/paper/1803.09941/) (2018)
- [Complexity and performance of an Augmented Lagrangian algorithm](https://mlsanity.com/paper/1907.02401/) (2019)
- [Neural Network Training as an Optimal Control Problem: An Augmented Lagrangian Approach](https://mlsanity.com/paper/2103.14343/) (2021)
- [Adaptive inexact fast augmented Lagrangian methods for constrained convex optimization](https://mlsanity.com/paper/1505.03175/) (2015)
- [Iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian method based on the classical Lagrangian function](https://mlsanity.com/paper/2008.00562/) (2020)
- [QPALM: A Proximal Augmented Lagrangian Method for Nonconvex Quadratic Programs](https://mlsanity.com/paper/2010.02653/) (2020)
- [Alpaqa: A matrix-free solver for nonlinear MPC and large-scale nonconvex optimization](https://mlsanity.com/paper/2112.02370/) (2021)
- [An adaptive superfast inexact proximal augmented Lagrangian method for smooth nonconvex composite optimization problems](https://mlsanity.com/paper/2207.11905/) (2022)
- [An inexact proximal augmented Lagrangian framework with arbitrary linearly convergent inner solver for composite convex optimization](https://mlsanity.com/paper/1909.09582/) (2019)

## Abstract (arXiv)

We propose an inexact proximal augmented Lagrangian method (P-ALM) for nonconvex structured optimization problems. The proposed method features an easily implementable rule not only for updating the penalty parameters, but also for adaptively tuning the proximal term. It allows the penalty parameter to grow rapidly in the early stages to speed up progress, while ameliorating the issue of ill-conditioning in later iterations, a well-known drawback of the traditional approach of linearly increasing the penalty parameters. A key element in our analysis lies in the observation that the augmented Lagrangian can be controlled effectively along the iterates, provided an initial feasible point is available. Our analysis, while simple, provides a new theoretical perspective about P-ALM and, as a by-product, results in similar convergence properties for its non-proximal variant, the classical augmented Lagrangian method (ALM). Numerical experiments, including convex and nonconvex problem instances, demonstrate the effectiveness of our approach.