The simplest examples where the simplex method cycles and conditions where EXPAND fails to prevent cycling

2000/12/22 by J. A. J. Hall, Hall, J. A. J., K. I. M. McKinnon +1
Computer Science · Mathematics · #Data Visualization and Analytics #FOS: Mathematics #Optimization and Control (math.OC) #math.OC

paper · pdf · doi:10.48550/arxiv.math/0012242

Submitted to Mathematical Programming

arxiv created 2000/12/22 · openalex publication_date 2000/12/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper introduces a class of linear programming examples which cause the simplex method to cycle indefinitely and which are the simplest possible examples showing this behaviour. The structure of examples from this class repeats after two iterations. Cycling is shown to occur for both the most negative reduced cost and steepest edge column selection criteria. In addition it is shown that the EXPAND anti-cycling procedure of Gill et al.is not guaranteed to prevent cycling.

Related

• Cycling problems in linear programming
• A generalization of the steepest-edge rule and its number of simplex iterations for a nondegenerate LP
• Linear Programming, the Simplex Algorithm and Simple Polytopes
• The Simplex Algorithm in Dimension Three
• The existence of a strongly polynomial time simplex algorithm for linear programs