The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S-moment problem and its dual counterpart: polynomials that are positive on S. It turns out that this methodology can also be used to solve problems with positivity constraints ‘f(x) ≥ 0 for all $\mathbf{x}\in S$ ’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.

中文翻译：

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Moment-SOS 层次结构：应用程序和相关主题


Moment-SOS 层次结构于 2000 年在优化中首次引入，它基于 S 矩问题及其对偶问题的理论：S 上为正的多项式。事实证明，该方法也可用于解决以下问题：正性约束 'f(x) ≥ 0 对于所有 $\mathbf{x}\in S$ ' 或 Borel 测量的线性约束。此类问题可以被视为广义矩问题（GMP）的具体实例，其在科学和工程各个领域的重要应用几乎是无穷无尽的。我们在优化中描述了这种方法，并在其他两个应用中进行了说明。最后，我们还介绍了 Christoffel 函数，并揭示了它与 Moment-SOS 层次结构和正多项式的联系。