# First-order methods for constrained convex programming based on linearized augmented Lagrangian function

arXiv: https://arxiv.org/abs/1711.08020 · HTML: https://mlsanity.com/paper/1711.08020/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2017.

## Built on by (most cited first, 9 of 9)

- [Training OOD Detectors in their Natural Habitats](https://mlsanity.com/paper/2202.03299/) (2022)
- [Stochastic Inexact Augmented Lagrangian Method for Nonconvex Expectation Constrained Optimization](https://mlsanity.com/paper/2212.09513/) (2022)
- [A single-loop SPIDER-type stochastic subgradient method for expectation-constrained nonconvex nonsmooth optimization](https://mlsanity.com/paper/2501.19214/) (2025)
- [Damped Proximal Augmented Lagrangian Method for weakly-Convex Problems with Convex Constraints](https://mlsanity.com/paper/2311.09065/) (2023)
- [GBM-based Bregman Proximal Algorithms for Constrained Learning](https://mlsanity.com/paper/2308.10767/) (2023)
- [Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions](https://mlsanity.com/paper/2502.19764/) (2025)
- [Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity](https://mlsanity.com/paper/2607.08954/) (2026)
- [Faster Accelerated First-order Methods for Convex Optimization with Strongly Convex Function Constraints](https://mlsanity.com/paper/2212.11143/) (2022)
- [Stochastic Trust-Region Methods for Over-parameterized Models](https://mlsanity.com/paper/2604.14017/) (2026)

## Abstract (arXiv)

First-order methods have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two first-order methods for constrained convex programs, for which the constraint set is represented by affine equations and smooth nonlinear inequalities. Both methods are based on the classic augmented Lagrangian function. They update the multipliers in the same way as the augmented Lagrangian method (ALM) but employ different primal variable updates. The first method, at each iteration, performs a single proximal gradient step to the primal variable, and the second method is a block update version of the first one. For the first method, we establish its global iterate convergence as well as global sublinear and local linear convergence, and for the second method, we show a global sublinear convergence result in expectation. Numerical experiments are carried out on the basis pursuit denoising and a convex quadratically constrained quadratic program to show the empirical performance of the proposed methods. Their numerical behaviors closely match the established theoretical results.