# Goldstein Stationarity in Lipschitz Constrained Optimization

arXiv: https://arxiv.org/abs/2310.03690 · HTML: https://mlsanity.com/paper/2310.03690/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Oct 2023.

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

- [First-Order Methods for Linearly Constrained Bilevel Optimization](https://mlsanity.com/paper/2406.12771/) (2024)
- [A single-loop SPIDER-type stochastic subgradient method for expectation-constrained nonconvex nonsmooth optimization](https://mlsanity.com/paper/2501.19214/) (2025)
- [Improved Sample Complexity for Private Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2410.05880/) (2024)
- [On the Hardness of Meaningful Local Guarantees in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2409.10323/) (2024)
- [The Exponential Price of Determinism in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2609.23837/) (2026)
- [Optimal Algorithms for Nonsmooth Nonconvex Stochastic Optimization with Constraints](https://mlsanity.com/paper/2610.13787/) (2026)

## Abstract (arXiv)

We prove the first convergence guarantees for a subgradient method minimizing a generic Lipschitz function over generic Lipschitz inequality constraints. No smoothness or convexity (or weak convexity) assumptions are made. Instead, we utilize a sequence of recent advances in Lipschitz unconstrained minimization, which showed convergence rates of O(1/δε3) towards reaching a "Goldstein" stationary point, that is, a point where an average of gradients sampled at most distance δ away has size at most ε. We generalize these prior techniques to handle functional constraints, proposing a subgradient-type method with similar O(1/δε3) guarantees on reaching a Goldstein Fritz-John or Goldstein KKT stationary point, depending on whether a certain Goldstein-style generalization of constraint qualification holds.