# A First-Order Primal-Dual Method for Nonconvex Constrained Optimization Based On the Augmented Lagrangian

arXiv: https://arxiv.org/abs/2007.12219 · HTML: https://mlsanity.com/paper/2007.12219/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jul 2020.

## Built on by (most cited first, 5 of 5)

- [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (2024)
- [Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity](https://mlsanity.com/paper/2607.08954/) (2026)
- [An Augmented Lagrangian Approach to Conically Constrained Non-monotone Variational Inequality Problems](https://mlsanity.com/paper/2306.01214/) (2023)
- [Differentially Private Neural Network Training Under the Hidden State Assumption](https://mlsanity.com/paper/2407.08233/) (2024)
- [DualHash: A Stochastic Primal-Dual Algorithm with Theoretical Guarantee for Deep Hashing](https://mlsanity.com/paper/2510.18218/) (2025)

## Abstract (arXiv)

Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics and data analytics. In this paper, based on the augmented Lagrangian function we introduce a flexible first-order primal-dual method, to be called nonconvex auxiliary problem principle of augmented Lagrangian (NAPP-AL), for solving a class of nonlinearly constrained nonconvex and nonsmooth optimization problems. We demonstrate that NAPP-AL converges to a stationary solution at the rate of o(1/√k), where k is the number of iterations. Moreover, under an additional error bound condition (to be called VP-EB in the paper), we further show that the convergence rate is in fact linear. Finally, we show that the famous Kurdyka- Lojasiewicz property and the metric subregularity imply the afore-mentioned VP-EB condition.