论文标题
通过多项式优化验证某些功能性不平等
Verification of some functional inequalities via polynomial optimization
论文作者
论文摘要
通过将lyapunov方法应用于部分微分方程（PDE）的动机，我们研究了$ f（i_1（u），\ ldots，i_k（u））\ geq 0 $ f $的功能不平等，其中$ f $是多项式，$ u $是$ u $的任何功能满意的函数，满足$ __1 $ $ $ $ i__1（u），$ i_1（u），其集成的功能是$ u $，其衍生物和集成变量的多项式。我们表明，这种功能性不平等可以加强到足够的多项式不平等中，从原则上可以使用标准技术进行多项式优化，可以通过半决赛编程来检查它们。这些足够的条件也可用于优化具有仿射依赖性可调参数的功能，同时确保其非阴性。我们的方法依赖于原始功能不平等的衡量理论提升，这既扩展了PDE分析的最新时刻放松策略，也扩展了整体功能的双重方法。
Motivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities of the form $f(I_1(u),\ldots,I_k(u))\geq 0$ where $f$ is a polynomial, $u$ is any function satisfying prescribed constraints, and $I_1(u),\ldots,I_k(u)$ are integral functionals whose integrands are polynomial in $u$, its derivatives, and the integration variable. We show that such functional inequalities can be strengthened into sufficient polynomial inequalities, which in principle can be checked via semidefinite programming using standard techniques for polynomial optimization. These sufficient conditions can be used also to optimize functionals with affine dependence on tunable parameters whilst ensuring their nonnegativity. Our approach relies on a measure-theoretic lifting of the original functional inequality, which extends both a recent moment relaxation strategy for PDE analysis and a dual approach to inequalities for integral functionals.
