The dual cone of sums of non-negative circuit polynomials

TL;DR

This paper characterizes the dual cone of sums of non-negative circuit polynomials, providing a new optimality criterion for polynomial optimization using these sums, advancing the theoretical understanding of their structure.

Contribution

The paper derives a representation of the dual cone of sums of non-negative circuit polynomials and introduces an optimality criterion for polynomial optimization.

Findings

Derived a representation of the dual cone $(C_{sonc}(\\mathcal{A}))^*$.

Provided an optimality criterion for sums of non-negative circuit polynomials.

Enhanced theoretical understanding of the structure of these cones.

Abstract

For a non-empty, finite subset $\mathcal{A} \subseteq \mathbb{N}_0^n$, denote by $C_{\text{sonc}}(\mathcal{A}) \in \mathbb{R}[x_1, \ldots, x_n]$ the cone of sums of non-negative circuit polynomials with support $\mathcal{A}$. We derive a representation of the dual cone $(C_{\text{sonc}}(\mathcal{A}))^*$ and deduce a resulting optimality criterion for the use of sums of non-negative circuit polynomials in polynomial optimization.