Mathematics Faculty Articles

Enumerating Extreme Points of the Polytopes of Stochastic Tensors: An Optimization Approach

Document Type

Article

Publication Date

4-2020

Publication Title

Optimization

Keywords

Birkhoff polytope, Birkhoff-von Neumann theorem, Extreme point, Line-stochastic tensor, Plane-stochastic tensor, Polytope, Tensor, Vertex

ISSN

0233-1934

Volume

69

Issue/No.

4

First Page

729

Last Page

741

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.

Comments

©2019 Informa UK Limited, trading as Taylor & Francis Group

Additional Comments

National Natural Science Foundatio nof China grant #s: 11571220, 11531001, 11271256; NSFC-ISF research program #: 11561141001; Montenegrin-Chinese Science and Technology Cooperation Project #: 3-12

DOI

10.1080/02331934.2019.1647198

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Peer Reviewed

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