计数随机张量多面体极值点的optimization方法
Enumerating extreme points of the polytopes of stochastic tensors: an optimization approac
摘要：
本文研究随机张量多面体的极值点。张量是指实数域上的多维数组。线随机张量是一种非负张量，其中每条线上的所有条目的总和（即，一个自由指数）等于1;平面随机张量是其中每个平面上的所有项的和（即，两个自由指数）等于1。在计算3阶和3维$n$的线随机张量和平面随机张量的多面体的极值点时，我们考虑了用线性最优化的方法，并给出了新的下界和上界.我们还研究了定义多面体的系数矩阵。
英文摘要：
This paper is concerned with the extreme points of the polytopes of stochastic tensors. By a tensor we mean a multi-dimensional array over the real number field. A line-stochastic tensor is a nonnegative tensor in which the sum of all entries on each line (i.e., one free index) is equal to 1; a plane-stochastic tensor is a nonnegative tensor in which the sum of all entries on each plane (i.e., two free indices) is equal to 1. In enumerating extreme points of the polytopes of line- and plane-stochastic tensors of order 3 and dimension $n$, we consider the approach by linear optimization and present new lower and upper bounds. We also study the coefficient matrices that define the polytopes.
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