# Iteration-complexity of a proximal augmented Lagrangian method for solving nonconvex composite optimization problems with nonlinear convex constraints

arXiv: https://arxiv.org/abs/2008.07080 · HTML: https://mlsanity.com/paper/2008.07080/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Aug 2020.

## Built on by (most cited first, 6 of 6)

- [Complexity of Single Loop Algorithms for Nonlinear Programming with Stochastic Objective and Constraints](https://mlsanity.com/paper/2311.00678/) (2023)
- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [Damped Proximal Augmented Lagrangian Method for weakly-Convex Problems with Convex Constraints](https://mlsanity.com/paper/2311.09065/) (2023)
- [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (2024)
- [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)
- [First-order Methods for Affinely Constrained Composite Non-convex Non-smooth Problems: Lower Complexity Bound and Near-optimal Methods](https://mlsanity.com/paper/2307.07605/) (2023)

## Abstract (arXiv)

This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear cal K-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone cal K. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the constrained problem in cal O(log(1/ρ)/ρ3) inner iterations, where ρ>0 is a given tolerance. Numerical experiments are also given to illustrate the computational efficiency of the proposed method.