# SONC Optimization and Exact Nonnegativity Certificates via Second-Order Cone Programming

arXiv: https://arxiv.org/abs/2012.07903 · HTML: https://mlsanity.com/paper/2012.07903/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2020.

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

- [Sparse Polynomial Optimization: Theory and Practice](https://mlsanity.com/paper/2208.11158/) (2022)
- [Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant](https://mlsanity.com/paper/2209.06175/) (2022)

## Abstract (arXiv)

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables.