# A Local-Linearly Convergent Algorithm for Nonconvex Equality-Constrained Optimization

arXiv: https://arxiv.org/abs/2608.12665 · HTML: https://mlsanity.com/paper/2608.12665/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 15 references that are in the map.
- First posted on arXiv 12 Aug 2026.

## Builds on (most cited first, 3 of 3)

- [Computing second-order points under equality constraints: revisiting Fletcher's augmented Lagrangian](https://mlsanity.com/paper/2204.01448/) (2022)
- [Implementing a smooth exact penalty function for equality-constrained nonlinear optimization](https://mlsanity.com/paper/1910.04300/) (2019)
- [Progressively Sampled Equality-Constrained Optimization](https://mlsanity.com/paper/2510.00417/) (2025)

## Abstract (arXiv)

For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.