Journal ArticleDOI
Pivot rules for linear programming: A survey on recent theoretical developments
Tamás Terlaky,Shuzhong Zhang +1 more
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TLDR
The various pivot rules of the simplex method and its variants that have been developed in the last two decades are discussed, starting from the appearance of the minimal index rule of Bland.Abstract:
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 this paper we discuss three classes of recently developed pivot rules for linear programming. The first and largest class is the class of essentially combinatorial pivot rules including minimal index type rules and recursive rules. These rules only use labeling and signs of the variables. The second class contains those pivot rules which can actually be considered as variants or generalizations or specializations of Lemke's method, and so they are closely related to parametric programming. The last class has the common feature that the rules all have close connections to certain interior point methods. Finally, we mention some open problems for future research.read more
Citations
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Journal ArticleDOI
A counterexample to the Hirsch conjecture
TL;DR: This paper presents the rst counterexample to the Hirsch Conjecture, obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.
Journal ArticleDOI
A counterexample to the Hirsch Conjecture
TL;DR: The first counterexample to the Hirsch conjecture has been presented in this paper, which is obtained from a 5-dimensional polytope with 48 and 86 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.
Posted Content
"Rotterdam econometrics": publications of the econometric institute 1956-2005
TL;DR: A list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005 can be found in this article.
Journal ArticleDOI
The use of the optimal partition in a linear programming solution for postoptimal analysis
TL;DR: Examples where the partition is what is needed or desired to perform the analysis and the importance of an optimal partition, induced by a strictly complementary solution, has surfaced in connection with the interior point method.
Journal ArticleDOI
Criss-cross methods: A fresh view on pivot algorithms
Komei Fukuda,Tamás Terlaky +1 more
TL;DR: A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms.
References
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Book
Linear Programming and Extensions
TL;DR: This classic book looks at a wealth of examples and develops linear programming methods for their solutions and begins by introducing the basic theory of linear inequalities and describes the powerful simplex method used to solve them.
Journal ArticleDOI
A new polynomial-time algorithm for linear programming
TL;DR: It is proved that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property: the ratio of the radius of the smallest sphere with center a′, containingP′ to theradius of the largest sphere withCenter a′ contained inP′ isO(n).
Book
The Linear Complementarity Problem
TL;DR: In this article, the authors present an overview of existing and multiplicity of degree theory and propose pivoting methods and iterative methods for degree analysis, including sensitivity and stability analysis.