# Third-Order Newton Methods with Semidefinite Programming

arXiv: https://arxiv.org/abs/2209.10051 · HTML: https://mlsanity.com/paper/2209.10051/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Sep 2022.

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

- [Higher-Order Newton Methods with Polynomial Work per Iteration](https://mlsanity.com/paper/2311.06374/) (2023)
- [On Approximate Computation of Critical Points](https://mlsanity.com/paper/2601.21917/) (2026)

## Abstract (arXiv)

We study a third-order Newton method in which each iteration is obtained by minimizing the third-order Taylor expansion of the objective via a semidefinite programming (SDP) subproblem. In contrast to classical Newton methods, which rely on quadratic approximations, our approach leverages higher-order information to better capture the local geometry of the objective. We establish cubic convergence under standard assumptions, including strong convexity and sufficiently accurate initialization. We also propose practical modifications to address infeasibility of the SDP subproblem, including regularization and alternative objective formulations.