Enumerating extreme points of the polytopes of stochastic tensors: an optimization approach. (2nd April 2020)

• Record Type: Journal Article
• Authors: Zhang, Fuzhen; Zhang, Xiao-Dong
• Is Part Of: Optimization. Volume 69:Number 4(2020)
• Page Start: 729
• Page End: 741
• Publication Date: 2020-04-02
• Subjects: Birkhoff polytope -- Birkhoff-von Neumann theorem -- extreme point -- line-stochastic tensor -- plane-stochastic tensor -- polytope -- tensor -- vertex; Primary 52B11 -- 15B51; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2019.1647198
• Languages: English
• ISSNs: 0233-1934

Abstract

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.