A Positivestellensatz for Conditional SAGE

Recently, conditional SAGE certificate has been proposed as a certificate of signomial positivity over a constrained set. In this article, we show that the conditional SAGE certificate is \textit{complete}. That is, for any signomial $f(\mathbf{x}) = \sum_{j=1}^{\ell}c_j \exp(\mathbf{A}_j\mathbf{x})$ defined by rational exponents that is positive over a compact convex set $\mathcal{X}$, there is $r \in \mathbb{Z}_+$ and a positive definite function $w(\mathbf{x})$ such that $w(\mathbf{x})^r f(\mathbf{x})$ is a conditional SAGE signomial with respect to the set $\mathcal{X}$. The completeness result is analogous to Positivestellensatz from algebraic geometry, which guarantees representation of positive polynomials with sum of squares polynomials. The result gives rise to a convergent hierarchy of lower bounds for constrained signomial optimization over \textit{arbitrary} compact convex set that is computable via the conditional SAGE certificate.