# The inexact power augmented Lagrangian method for constrained nonconvex optimization

arXiv: https://arxiv.org/abs/2410.20153 · HTML: https://mlsanity.com/paper/2410.20153/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 60 references that are in the map.
- First posted on arXiv 26 Oct 2024.

## Builds on (most cited first, 28 of 28)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Adaptive Gradient Descent without Descent](https://mlsanity.com/paper/1910.09529/) (2019)
- [Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming](https://mlsanity.com/paper/1310.3787/) (2013)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints](https://mlsanity.com/paper/1906.11357/) (2019)
- [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)
- [Clustering subgaussian mixtures by semidefinite programming](https://mlsanity.com/paper/1602.06612/) (2016)
- [Constrained Optimization to Train Neural Networks on Critical and Under-Represented Classes](https://mlsanity.com/paper/2102.12894/) (2021)
- [Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient](https://mlsanity.com/paper/2301.04431/) (2023)
- [A simple uniformly optimal method without line search for convex optimization](https://mlsanity.com/paper/2310.10082/) (2023)
- [Complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs](https://mlsanity.com/paper/1802.03504/) (2018)
- [Bundle-Level Type Methods Uniformly Optimal for Smooth and Nonsmooth Convex Optimization](https://mlsanity.com/paper/1309.5547/) (2013)
- [A Single-Loop Gradient Descent and Perturbed Ascent Algorithm for Nonconvex Functional Constrained Optimization](https://mlsanity.com/paper/2207.05650/) (2022)
- [Efficient Algorithms for Large-scale Generalized Eigenvector Computation and Canonical Correlation Analysis](https://mlsanity.com/paper/1604.03930/) (2016)
- [On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization](https://mlsanity.com/paper/1906.05622/) (2019)
- [On Optimal Universal First-Order Methods for Minimizing Heterogeneous Sums](https://mlsanity.com/paper/2208.08549/) (2022)
- [Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence](https://mlsanity.com/paper/1801.03013/) (2018)
- [Smoothness parameter of power of Euclidean norm](https://mlsanity.com/paper/1907.12346/) (2019)
- [Neural Network Training as an Optimal Control Problem: An Augmented Lagrangian Approach](https://mlsanity.com/paper/2103.14343/) (2021)
- [An Inexact Augmented Lagrangian Algorithm for Training Leaky ReLU Neural Network with Group Sparsity](https://mlsanity.com/paper/2205.05428/) (2022)
- [Generalized Uniformly Optimal Methods for Nonlinear Programming](https://mlsanity.com/paper/1508.07384/) (2015)
- [QPALM: A Proximal Augmented Lagrangian Method for Nonconvex Quadratic Programs](https://mlsanity.com/paper/2010.02653/) (2020)
- [Adaptive proximal gradient methods are universal without approximation](https://mlsanity.com/paper/2402.06271/) (2024)
- [High-order methods beyond the classical complexity bounds, I: inexact high-order proximal-point methods](https://mlsanity.com/paper/2107.05958/) (2021)
- [High-order methods beyond the classical complexity bounds, II: inexact high-order proximal-point methods with segment search](https://mlsanity.com/paper/2109.12303/) (2021)
- [Global Convergence Analysis of the Power Proximal Point and Augmented Lagrangian Method](https://mlsanity.com/paper/2312.12205/) (2023)
- [Computational Complexity of Inexact Gradient Augmented Lagrangian Methods: Application to Constrained MPC](https://mlsanity.com/paper/1302.4355/) (2013)

## Built on by (most cited first, 1 of 1)

- [A proximal augmented Lagrangian method for nonconvex optimization with equality and inequality constraints](https://mlsanity.com/paper/2509.02894/) (2025)

## Abstract (arXiv)

This work introduces an unconventional inexact augmented Lagrangian method where the augmenting term is a Euclidean norm raised to a power between one and two. The proposed algorithm is applicable to a broad class of constrained nonconvex minimization problems that involve nonlinear equality constraints. In a first part of this work, we conduct a full complexity analysis of the method under a mild regularity condition, leveraging an accelerated first-order algorithm for solving the H"older-smooth subproblems. Interestingly, this worst-case result indicates that using lower powers for the augmenting term leads to faster constraint satisfaction, albeit with a slower decrease of the dual residual. Notably, our analysis does not assume boundedness of the iterates. Thereafter, we present an inexact proximal point method for solving the weakly-convex and H"older-smooth subproblems, and demonstrate that the combined scheme attains an improved rate that reduces to the best-known convergence rate whenever the augmenting term is a classical squared Euclidean norm. Different augmenting terms, involving a lower power, further improve the primal complexity at the cost of the dual complexity. Finally, numerical experiments validate the practical performance of unconventional augmenting terms.