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

  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},
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… 

The Polyhedral Geometry of Pivot Rules and Monotone Paths

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

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

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

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

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

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.

On the Simplex method for 0/1 polytopes

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

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

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.



What is the Worst Case Behavior of the Simplex Algorithm

The purpose of this paper is to explain the Klee-Minty and Jeroslow constructions, show how they can be modified to be pathological with small integral coefficients, and then suggest a 'least entered' pivot rule which forces an improving column to be entered before any other column is entered for the second time.

A Monotonic Build-Up Simplex Algorithm for Linear Programming

A new simplex pivot rule is devised which produces a sequence of pivots such that ultimately the originally chosen nonbasic variable enters the basis, and all reduced costs which were originally nonnegative remain nonnegative.

The pivot and probe algorithm for solving a linear program

Computational experience is presented indicating that time savings of 50–80% over the simplex method can be obtained by this method, which is called PAPA, the Pivot and Probe Algorithm.

Resolving degeneracy in combinatorial linear programs: Steepest edge, steepest ascent, and parametric ascent

Algorithms described here have the advantage that they choose the pivot element without explicit knowledge of the set of all active constraints at a point of degeneracy, thus making them attractive in combinatorial settings where the linear program is represented implicitly.

Parametric linear programming and anti-cycling pivoting rules

This work develops anticycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule, so any variable that ties for the ratio rule can leave the basis.

Least-Index Resolution of Degeneracy in Linear Complementarity Problems with Sufficient Matrices

It is shown that for circling to occur in the principal pivoting method, the matrix must have order at least four, and for Lemke's algorithm it must be at least three, and examples are given showing that these bounds are sharp.

The d-Step Conjecture and Its Relatives

This report summarizes what is known about the d-step conjecture and its relatives and includes the first example of a polytope that is not vertex-decomposable, showing that a certain natural approach to the conjecture will not work.

A new algorithm for quadratic programming

Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems

  • M. Todd
  • Computer Science, Mathematics
    Math. Program.
  • 1986
We show that a particular pivoting algorithm, which we call the lexicographic Lemke algorithm, takes an expected number of steps that is bounded by a quadratic inn, when applied to a random linear