# On representing the positive semidefinite cone using the second-order cone

arXiv: https://arxiv.org/abs/1610.04901 · HTML: https://mlsanity.com/paper/1610.04901/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2016.

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

- [Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family](https://mlsanity.com/paper/2508.14369/) (2025)
- [Multiplicative updates for symmetric-cone factorizations](https://mlsanity.com/paper/2108.00740/) (2021)

## Abstract (arXiv)

The second-order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity. Second-order cone formulations can also be solved more efficiently than semidefinite programming in general. We consider the following question, posed by Lewis and Glineur, Parrilo, Saunderson: is it possible to express the general positive semidefinite cone using second-order cones? We provide a negative answer to this question and show that the 3x3 positive semidefinite cone does not admit any second-order cone representation. Our proof relies on exhibiting a sequence of submatrices of the slack matrix of the 3x3 positive semidefinite cone whose "second-order cone rank" grows to infinity. We also discuss the possibility of representing certain slices of the 3x3 positive semidefinite cone using the second-order cone.