# Worst-case iteration bounds for log barrier methods on problems with nonconvex constraints

arXiv: https://arxiv.org/abs/1807.00404 · HTML: https://mlsanity.com/paper/1807.00404/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2018.

## Built on by (most cited first, 2 of 2)

- [A learning-based approach to stochastic optimal control under reach-avoid constraint](https://mlsanity.com/paper/2412.16561/) (2024)
- [Nash Equilibria in Games with Playerwise Concave Coupling Constraints: Existence and Computation](https://mlsanity.com/paper/2509.14032/) (2025)

## Abstract (arXiv)

Interior point methods (IPMs) that handle nonconvex constraints such as IPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs in the setting where the objective and constraints are thrice differentiable, and have Lipschitz first and second derivatives on the feasible region. We provide an IPM that, starting from a strictly feasible point, finds a μ-approximate Fritz John point by solving O( μ-7/4) trust-region subproblems. For IPMs that handle nonlinear constraints, this result represents the first iteration bound with a polynomial dependence on 1/μ. We also show how to use our method to find scaled-KKT points starting from an infeasible solution and improve on existing complexity bounds.