Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds for the number of basic feasible solutions generated by the simplex method with the largest distance rule

Tomonari Kitahara

Abstract

In this paper, we analyze the simplex method with the largest distance rule and derive upper bounds on the number of different basic feasible solutions generated. The pivoting rule was proposed by Pan [10], and in some cases, it was reported to be more efficient than the renowned steepest edge rule. We show that the analytical framework developed by Kitahara and Mizuno can be extended to this rule, despite its structural differences from previously studied pivoting rules. The resulting bounds involve a geometric parameter $β$ determined by the column norms of the constraint matrix. In addition, our analysis does not require a nondegeneracy assumption.

Bounds for the number of basic feasible solutions generated by the simplex method with the largest distance rule

Abstract

In this paper, we analyze the simplex method with the largest distance rule and derive upper bounds on the number of different basic feasible solutions generated. The pivoting rule was proposed by Pan [10], and in some cases, it was reported to be more efficient than the renowned steepest edge rule. We show that the analytical framework developed by Kitahara and Mizuno can be extended to this rule, despite its structural differences from previously studied pivoting rules. The resulting bounds involve a geometric parameter determined by the column norms of the constraint matrix. In addition, our analysis does not require a nondegeneracy assumption.

Paper Structure

This paper contains 15 sections, 6 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main contributions
  4. Notations and the structure of the paper
  5. Preliminaries and existing research
  6. Linear programming and the simplex method
  7. Pivoting rules
  8. Parameters
  9. Assumptions
  10. Previous research
  11. Analysis of the largest distance rule
  12. Technical results
  13. A bound using the objective function
  14. A bound independent of the objective function
  15. Conclusion

Key Result

Lemma 3.1

Let $x^t$ be the $t$-th solution generated by the simplex method with any pivoting rule, and let $B^t$ and $N^t$ be the basis and nonbasis, respectively, corresponding to $x^t$. Set $\Delta_{d,t} = -\min\{\bar{c}_j|~j\in N^t\}$. Then, we have

Theorems & Definitions (9)

  • Lemma 3.1: Kitahara and Mizuno km2013
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.3: Kitahara and Mizuno km2013
  • Theorem 3.2
  • proof
  • Corollary 3.1