# Pivot rules for linear programming: A survey on recent theoretical developments

@article{Terlaky1993PivotRF, title={Pivot rules for linear programming: A survey on recent theoretical developments}, author={Tam{\'a}s Terlaky and Shuzhong Zhang}, journal={Annals of Operations Research}, year={1993}, volume={46-47}, pages={203-233} }

The purpose of this paper is to discuss the various pivot rules of the simplex method and its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with finiteness properties of simplex type pivot rules. Well known classical results concerning the simplex method are not considered in this survey, but the connection between the new pivot methods and the classical ones, if there is any, is discussed.In…

## 138 Citations

### The Polyhedral Geometry of Pivot Rules and Monotone Paths

- Mathematics
- 2022

Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are…

### New Optimal Pivot Rule for the Simplex Algorithm

- Computer Science
- 2016

A pivot rule is proposed that can reduce the number of such iterations over the Dantzig’s pivot rule and prevent cycling in the simplex algorithm and leads to an optimal improvement of the objective function at each iteration.

### Computing and proving with pivots

- Mathematics, Computer ScienceRAIRO Oper. Res.
- 2013

This present paper is a survey on algorithms in operations research and discrete mathematics using pivots, and gives also examples where this principle allows not only to compute but also to prove some theorems in a constructive way.

### Pivot Rules for Circuit-Augmentation Algorithms in Linear Optimization

- Computer ScienceSIAM Journal on Optimization
- 2022

It is proved that (i) computing the shortest monotone path to an optimal solution on the 1-skeleton of a polytope is NP-hard, and hard to approximate within a factor better than 2, and (ii) for 0/1 polytopes, a monot one path of polynomial length can be constructed using steepest improving edges.

### A double-pivot simplex algorithm and its upper bounds of the iteration numbers

- Mathematics
- 2019

In this paper, a double-pivot simplex method is proposed. Two upper bounds of iteration numbers are derived. Applying one of the bounds to some special linear programming (LP) problems, such as LP…

### Anstreicher–Terlaky type monotonic simplex algorithms for linear feasibility problems

- Mathematics, Computer ScienceOptim. Methods Softw.
- 2007

A new monotonic build-up (MBU) simplex algorithm for linear feasibility problems and a new recursive procedure to handle strongly degenerate problems as well are constructed.

### The s-monotone index selection rules for pivot algorithms of linear programming

- Computer Science, MathematicsEur. J. Oper. Res.
- 2012

### On the Simplex method for 0/1 polytopes

- Mathematics, Computer Science
- 2021

We present new pivot rules for the Simplex method for LPs over 0/1 polytopes. We show that the number of non-degenerate steps taken using these rules is strongly polynomial and even linear in the…

### Computational aspects of simplex and MBU-simplex algorithms using different anti-cycling pivot rules

- Computer Science
- 2014

The practical benefit of the flexibility of these anti-cycling pivot rules is evaluated using public benchmark LP test sets and the results provide numerical evidence that the MBU-simplex algorithm is a viable alternative to the traditional simplex algorithm.

### Positive Edge: A Pricing Criterion for the Identification of Non-Degenerate Simplex Pivots

- Computer Science
- 2010

A simple algorithm is designed using two external procedures: one identifies variables that allow for non-degenerate pivots while the other identifies variables with negative reduced cost that are sent to the primal simplex algorithm of cplex.

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