Enumerating Extreme Points of the Polytopes of Stochastic Tensors: An Optimization Approach
Birkhoff polytope, Birkhoff-von Neumann theorem, Extreme point, Line-stochastic tensor, Plane-stochastic tensor, Polytope, Tensor, Vertex
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.
Zhang, Fuzhen and Zhang, Xiao-Dong, "Enumerating Extreme Points of the Polytopes of Stochastic Tensors: An Optimization Approach" (2020). Mathematics Faculty Articles. 293.