# A unified framework of SAGE and SONC polynomials and its duality theory

arXiv: https://arxiv.org/abs/1903.08966 · HTML: https://mlsanity.com/paper/1903.08966/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Mar 2019.

## Built on by (most cited first, 1 of 1)

- [Sparse Polynomial Optimization: Theory and Practice](https://mlsanity.com/paper/2208.11158/) (2022)

## Abstract (arXiv)

We introduce and study a cone which consists of a class of generalized polynomial functions and which provides a common framework for recent non-negativity certificates of polynomials in sparse settings. Specifically, this S-cone generalizes and unifies sums of arithmetic-geometric mean exponentials (SAGE) and sums of non-negative circuit polynomials (SONC). We provide a comprehensive characterization of the dual cone of the S-cone, which even for its specializations provides novel and projection-free descriptions. As applications of this result, we give an exact characterization of the extreme rays of the S-cone and thus also of its specializations, and we provide a subclass of functions for which non-negativity coincides with membership in the S-cone. Moreover, we derive from the duality theory an approximation result of non-negative univariate polynomials and show that a SONC analogue of Putinar's Positivstellensatz does not exist even in the univariate case.