论文标题
最小化理性函数：通过推送措施的近似结构层次结构
Minimizing rational functions: a hierarchy of approximations via pushforward measures
论文作者
论文摘要
本文涉及最大程度地减少一组紧凑的高维度的理性功能。我们的方法依赖于第二个拉赛尔的层次结构（也称为上限层次结构），以在推送措施上提出，以便在较小维度的空间中工作。我们表明，在一般情况下，最小值可以按照上面的层次结构近似，并且具有半标准程序问题的层次结构，或者在单个分数的特定情况下，以及具有广义特征值问题的层次结构。 我们从数值上说明了使用推送度量而不是标准上限层次结构的潜力。我们认为，这种潜力应该是一个很大的动力，以研究一个相关的具有挑战性的问题。即，相对于Lebesgue或HAAR度量，将给定多项式的任意功率集成在简单的集合（例如单位盒或单位球体）上。
This paper is concerned with minimizing a sum of rational functions over a compact set of high-dimension. Our approach relies on the second Lasserre's hierarchy (also known as the upper bounds hierarchy) formulated on the pushforward measure in order to work in a space of smaller dimension. We show that in the general case the minimum can be approximated as closely as desired from above with a hierarchy of semidefinite programs problems or, in the particular case of a single fraction, with a hierarchy of generalized eigenvalue problems. We numerically illustrate the potential of using the pushforward measure rather than the standard upper bounds hierarchy. In our opinion, this potential should be a strong incentive to investigate a related challenging problem interesting in its own; namely integrating an arbitrary power of a given polynomial on a simple set (e.g., unit box or unit sphere) with respect to Lebesgue or Haar measure.
