论文标题
Moment Sos层次结构和ChristOffel-Darboux内核
The Moment-SOS hierarchy and the Christoffel-Darboux kernel
论文作者
论文摘要
我们认为多项式在紧凑型集合B上的全局最小化。我们表明，瞬间SOS层次结构的每个步骤都有一个很好而简单的解释，可以补充通常的解释。即，它在正常基础的基础上计算多项式的系数（b，$ $ $），其中$μ$是一种任意的参考措施，其恰好的支持是B的B。在某种程度上，一定程度的签名措施是一定的。全局最小化和最佳（签名）密度的多项式与ChristOffel-Darboux（CD）内核和与$μ$相关的ChristOffel功能有关。与计算正密度的上限的层次结构相反，可以完全作为对多项式（签名）密度的集成而实现的全局最佳，因为CD-kernel是繁殖的核，因此可以模仿狄拉克的量度（只要有一定的时刻，就可以模仿狄拉克的措施）。
We consider the global minimization of a polynomial on a compact set B. We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of L 2 (B, $μ$) where $μ$ is an arbitrary reference measure whose support is exactly B. The resulting polynomial is a certain density (with respect to $μ$) of some signed measure on B. When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel-Darboux (CD) kernel and the Christoffel function associated with $μ$. In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).
