Short communication: A note on cycling LP examples with permutation structure

Authors:
Peter Zörnig
Affiliations:
Departamento de Estatística, Instituto de Ciências Exatas, Universidade de Brasília, 70910-900 Brasília-DF, Brazil
Venue:
Computers and Operations Research
Year:
2008

Citing 8
Cited 0

Matrix analysis

Matrix analysis
Classroom Note: Hoffman's Circle Untangled

SIAM Review
Accuracy and Stability of Numerical Algorithms

Accuracy and Stability of Numerical Algorithms
Cycling in linear programming problems

Computers and Operations Research
The simplest examples where the simplex method cycles and conditions where expand fails to prevent cycling

Mathematical Programming: Series A and B
Systematic construction of examples for cycling in the simplex method

Computers and Operations Research
On the Newton Method for the Matrix Pth Root

SIAM Journal on Matrix Analysis and Applications
On Hoffman's celebrated cycling LP example

Computers and Operations Research

Quantified Score

Hi-index	0.01

Visualization

Abstract

We study a particular class of cycling examples for the simplex algorithm which is characterized by a permutation structure, i.e. after a few iterations the simplex tableau is a column permutation of the initial tableau. Developing some matrix theory, among others, we formulate necessary and sufficient conditions, characterizing cycling examples with this structure and give some numerical examples. Hoffman's cycling example [Hoffman AJ. Cycling in the simplex algorithm. Report 2974, Washington DC: National Bureau of Standards; 1953] turns out to be a special member of this class. The answers on some related questions, studied very recently in Guerrero-Garcia and Santos-Palomo [On Hoffman's celebrated cycling LP example. Computers & Operations Research, to appear] can be easily obtained in terms of the general framework.