A Positivstellensatz for Conditional SAGE Signomials

• URL: http://arxiv.org/abs/2003.03731v4
• Date: Sat, 24 Oct 2020 11:18:26 GMT
• Title: A Positivstellensatz for Conditional SAGE Signomials
• Abstract summary: We show that the conditional SAGE certificate is $textitcomplete$. $p in mathbbZ_+$ and a specific positive definite function $w(mathbfx)$ may be verified by the certificate.
• Score: 38.44747913332153
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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