# Convexity Certificates from Hessians

arXiv: https://arxiv.org/abs/2210.10430 · HTML: https://mlsanity.com/paper/2210.10430/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 28 references that are in the map.
- First posted on arXiv 19 Oct 2022.

## Builds on (most cited first, 7 of 7)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- ["Convex Until Proven Guilty": Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions](https://mlsanity.com/paper/1705.02766/) (2017)
- [GENO -- GENeric Optimization for Classical Machine Learning](https://mlsanity.com/paper/1905.13587/) (2019)
- [Convex Optimization in Julia](https://mlsanity.com/paper/1410.4821/) (2014)
- [Optimization for Classical Machine Learning Problems on the GPU](https://mlsanity.com/paper/2203.16340/) (2022)
- [On the Complexity of Detecting Convexity over a Box](https://mlsanity.com/paper/1806.06173/) (2018)

## Abstract (arXiv)

The Hessian of a differentiable convex function is positive semidefinite. Therefore, checking the Hessian of a given function is a natural approach to certify convexity. However, implementing this approach is not straightforward since it requires a representation of the Hessian that allows its analysis. Here, we implement this approach for a class of functions that is rich enough to support classical machine learning. For this class of functions, it was recently shown how to compute computational graphs of their Hessians. We show how to check these graphs for positive semidefiniteness. We compare our implementation of the Hessian approach with the well-established disciplined convex programming (DCP) approach and prove that the Hessian approach is at least as powerful as the DCP approach for differentiable functions. Furthermore, we show for a state-of-the-art implementation of the DCP approach that, for differentiable functions, the Hessian approach is actually more powerful. That is, it can certify the convexity of a larger class of differentiable functions.