# Computing second-order points under equality constraints: revisiting Fletcher's augmented Lagrangian

arXiv: https://arxiv.org/abs/2204.01448 · HTML: https://mlsanity.com/paper/2204.01448/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2022.

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

- [Optimization without Retraction on the Random Generalized Stiefel Manifold](https://mlsanity.com/paper/2405.01702/) (2024)
- [A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees](https://mlsanity.com/paper/2207.05697/) (2022)
- [A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization](https://mlsanity.com/paper/2301.04204/) (2023)
- [Progressively Sampled Equality-Constrained Optimization](https://mlsanity.com/paper/2510.00417/) (2025)
- [A second-order method landing on the Stiefel manifold via Newtonunicodex2013Schulz iteration](https://mlsanity.com/paper/2605.02838/) (2026)
- [A Local-Linearly Convergent Algorithm for Nonconvex Equality-Constrained Optimization](https://mlsanity.com/paper/2608.12665/) (2026)

## Abstract (arXiv)

We address the problem of minimizing a smooth function under smooth equality constraints. Under regularity assumptions on these constraints, we propose a notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization. Using a smooth exact penalty function known as Fletcher's augmented Lagrangian, we propose an algorithm to minimize the penalized cost function which reaches varepsilon-approximate second-order critical points of the original optimization problem in at most O(varepsilon-3) iterations. This improves on current best theoretical bounds. Along the way, we show new properties of Fletcher's augmented Lagrangian, which may be of independent interest.