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… 

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References

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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.

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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.

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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.

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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.

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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.

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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
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