A Positivstellensatz for Conditional SAGE Signomials
This provides a theoretical foundation for optimization problems in applied mathematics and engineering, though it is incremental as it extends prior SAGE results.
The authors proved that the conditional SAGE certificate is complete for verifying positivity of signomials over compact convex sets, establishing a convergent hierarchy of lower bounds for signomial optimization.
Recently, the conditional SAGE certificate has been proposed as a sufficient condition for signomial positivity over a convex 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 $p \in \mathbb{Z}_+$ and a specific positive definite function $w(\mathbf{x})$ such that $w(\mathbf{x})^p f(\mathbf{x})$ may be verified by the conditional SAGE certificate. The completeness result is analogous to Positivstellensatz results 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 an $\textit{arbitrary}$ compact convex set that is computable via the conditional SAGE certificate.