条件SAGE符号的正性定理
A Positivstellensatz for Conditional SAGE Signomials
摘要：
最近，有条件SAGE证书已被提出作为凸集上符号正性的充分条件。在本文中，我们展示了条件SAGE证书是$\textit{complete}$。也就是说，对于由在紧致凸集$\mathcal{X}$上为正的有理指数定义的任何符号$f(\mathbf{x}) = \sum_{j=1}^{\ell}c_j \exp(\mathbf{A}_j\mathbf{x})$，存在$p \in \mathbb{Z}_+$和特定的正定函数$w(\mathbf{x})$，使得$w(\mathbf{x})^p f(\mathbf{x})$可以由条件SAGE证书验证。完备性结果类似于代数几何中的Positivstellenestructive结果，它保证了正多项式的平方和多项式的表示。结果给出了一个收敛的层次结构的下界约束符号优化的$\textit{arbitrary}$紧凑的凸集，可计算的条件SAGE证书。
英文摘要：
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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