# Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors

@inproceedings{Zhang2021ComparisonOT, title={Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors}, author={Fuzhen Zhang and Xiaodong Zhang}, year={2021} }

We call a real multi-dimensional array a tensor for short. In enumerating vertices of the polytopes of stochastic tensors, different approaches have been used: (1) Combinatorial method via Latin squares; (2) Analytic (topological) approach by using hyperplanes; (3) Computational geometry (polytope theory) approach; and (4) Optimization (linear programming) approach. As all these approaches are worthy of consideration and investigation in the enumeration problem, various bounds have been…

## References

SHOWING 1-10 OF 30 REFERENCES

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

- MathematicsOptimization
- 2019

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…

On the number of vertices of the stochastic tensor polytope

- Mathematics
- 2017

Abstract This paper studies lower and upper bounds for the number of vertices of the polytope of stochastic tensors (i.e. triply stochastic arrays of dimension n). By using known results on polytopes…

Polytopes of Stochastic Tensors

- Mathematics
- 2016

Considering $n\times n\times n$ stochastic tensors $(a_{ijk})$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope ($\Omega_{n}$) of all…

Birkhoff-von Neumann theorem and decomposition for doubly stochastic tensors

- Mathematics
- 2019

Abstract The well-known Birkhoff-von Neumann theorem states that a doubly stochastic matrix is a convex combination of permutation matrices. In this paper, we present a new concept for doubly…

Tensor eigenvalues and their applications (advances in mechanics and mathematics, band 39)

- Mathematics
- 2018

The Permanent Functions of Tensors

- MathematicsActa Mathematica Vietnamica
- 2018

By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the…

Characterization of Extreme Points of Multi-Stochastic Tensors

- Computer Science, MathematicsComput. Methods Appl. Math.
- 2016

An algorithm to search the convex combination of extreme points for an arbitrary given multi-stochastic tensor is developed and a new approach for the partially filled square problem under the framework of multi- Stochastic Tensors is given.

Theory and Computation of Tensors: Multi-Dimensional Arrays

- Computer Science
- 2016

Theory and Computation of Tensors: Multi-Dimensional Arrays investigates theories and computations of tensors to broaden perspectives on matrices and discusses how to extend numerical linear algebra to numerical multi-linear algebra.