标题： 任意维数的多项式优化通过德芬内蒂定理 显示英文标题

标题： Any-Dimensional Polynomial Optimization via de Finetti Theorems

摘要： 多项式优化问题经常出现在由维度索引的序列中，产生所有问题中最优值的界限是有意义的。 示例包括证明在所有大小的向量和图上成立的对称函数或图同态密度之间的不等式，以及计算作为玩家数量增长的游戏极限的均场博弈的值。 在本文中，我们使用表示稳定性理论来研究这种任意维的多项式问题，并开发了一个系统框架，以有限维多项式优化问题的形式生成其极限最优值的界限层次。 我们的界限以明确的速率收敛，并且作为涉及不同方式投影到彼此的随机数组序列的新de Finetti型定理的结果出现。 这些定理的证明基于将概率结果应用于某些范畴的表示。 我们将我们的框架应用于产生来自多个应用领域的问题的新界限，例如均场博弈、极图论和对称函数理论，并通过数值实验说明我们的方法。 显示英文摘要

摘要： Polynomial optimization problems often arise in sequences indexed by dimension, and it is of interest to produce bounds on the optimal values of all problems in the sequence. Examples include certifying inequalities between symmetric functions or graph homomorphism densities that hold over vectors and graphs of all sizes, and computing the value of mean-field games viewed as limits of games with a growing number of players. In this paper, we study such any-dimensional polynomial problems using the theory of representation stability, and we develop a systematic framework to produce hierarchies of bounds for their limiting optimal values in terms of finite-dimensional polynomial optimization problems. Our bounds converge at explicit rates, and they follow as a consequence of new de Finetti-type theorems pertaining to sequences of random arrays projecting onto each other in different ways. The proofs of these theorems are based on applying results from probability to representations of certain categories. We apply our framework to produce new bounds on problems arising in a number of application domains such as mean-field games, extremal graph theory, and symmetric function theory, and we illustrate our methods via numerical experiments.