Enumerating extreme points of the polytopes of stochastic tensors: an optimization approac

Abstract

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.