论文标题
有条件的鼠尾草障碍的Potitivstellensatz
A Positivstellensatz for Conditional SAGE Signomials
论文作者
论文摘要
最近，有条件的鼠尾草证书被认为是在凸组集合中具有障碍阳性的足够条件。在本文中，我们表明有条件的鼠尾草证书为$ \ textit {poutter} $。也就是说，对于任何诱人的$ f（\ Mathbf {x}）= \ sum_ {j = 1}^{\ ell} c_j \ exp（\ Mathbf {\ Mathbf {a} _J \ Mathbf {x}} $ convex $ p} $ p} $ p} $ \ s $ paste $ \ math $ {x） \ Mathbb {Z} _+$和一个特定的正定义函数$ W（\ Mathbf {X}）$，使得$ W（\ MathBf {X}）^P F（\ MathBf {X}）$可以通过条件的Sage证书来验证。完整性结果类似于代数几何形状产生的stitivstellensatz，该几何形状保证了用正方形总和多项式的阳性多项式表示。该结果产生了下限的收敛层次结构，以在$ \ textIt {nutionary} $ compact convex集上进行约束尖端优化，该集合可通过有条件的SAGE证书进行计算。
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.
