We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by computing the smallest (rest. largest) eigenvalue of a hierarchy of (r x r) tri-diagonal univariate moment matrices of increasing size. Equivalently it reduces to computing the smallest (resp. largest) root of a certain univariate degree-r orthonormal polynomial. This provides a strong connection between the fields of optimization, orthogonal polynomials, numerical analysis and linear algebra, via asymptotic spectral analysis of tri-diagonal symmetric matrices.

中文翻译：

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连接优化与三对角矩阵的谱分析

我们表明，通过计算 (rxr) 三对角线的层次结构的最小（其余最大）特征值，可以从上面（从下面）近似计算紧凑集上连续函数的全局最小值（相应最大值）大小增加的单变量矩矩阵。等效地，它简化为计算某个单变量阶数-r 正交多项式的最小（或最大）根。通过三对角对称矩阵的渐近谱分析，这在优化、正交多项式、数值分析和线性代数领域之间提供了强大的联系。