关于随机多项式的极小值个数
数值分析
2010-07-12 v2 最优化与控制
概率论
摘要
对于次数为 d、变量至多 n 个的正态随机多项式，我们给出其临界点个数的 O(d ^((n+1)/2)) 上界。利用大随机矩阵谱值的大偏差原理，我们得到此类多项式极小值个数的界 O(exp(-beta n^2 + (n/2) log (d-1)))(beta 为独立于 n 和 d 的正常数)。这证明了大多数固定次数的正态随机多项式只有鞍点。最后，我们用超几何函数给出了随机单变量多项式极大值(相应地极小值)个数的闭式表达。
引用
@article{arxiv.math/0702360,
title = {On the number of minima of a random polynomial},
author = {Jean-Pierre Dedieu and Gregorio Malajovich},
journal= {arXiv preprint arXiv:math/0702360},
year = {2010}
}
备注
22 pages. We learned since the first version that the probability that a matrix in GOE(n) is positive definite is known. This follows from the theory of large deviations (reference in the paper). Therefore, we can now state a precise upper bound (Theorem 2) for the number of minima of a random polynomial, instead of a bound depending on that probability