# Iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian method based on the classical Lagrangian function

arXiv: https://arxiv.org/abs/2008.00562 · HTML: https://mlsanity.com/paper/2008.00562/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Aug 2020.

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

- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [Stochastic Smoothed Primal-Dual Algorithms for Nonconvex Optimization with Linear Inequality Constraints](https://mlsanity.com/paper/2504.07607/) (2025)
- [Federated Learning with Convex Global and Local Constraints](https://mlsanity.com/paper/2310.10117/) (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)
- [A proximal augmented Lagrangian method for nonconvex optimization with equality and inequality constraints](https://mlsanity.com/paper/2509.02894/) (2025)

## Abstract (arXiv)

This paper establishes the iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian (IAIPAL) method for solving linearly-constrained smooth nonconvex composite optimization problems that is based on the classical augmented Lagrangian (AL) function. More specifically, each IAIPAL iteration consists of inexactly solving a proximal AL subproblem by an accelerated composite gradient (ACG) method followed by a classical Lagrange multiplier update. Under the assumption that the domain of the composite function is bounded and the problem has a Slater point, it is shown that IAIPAL generates an approximate stationary solution in cal O(varepsilon-5/2log2 varepsilon-1) ACG iterations where varepsilon>0 is a tolerance for both stationarity and feasibility. Moreover, the above bound is derived without assuming that the initial point is feasible. Finally, numerical results are presented to demonstrate the strong practical performance of IAIPAL.