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CrissCross Pivoting Rules
"... . Assuming that the reader is familiar with both the primal and dual simplex methods, Zionts' crisscross method can easily be explained. ffl It can be initialized by any, possibly both primal and dual infeasible basis . If the basis is optimal, we are done. If the basis is not optimal , th ..."
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. Assuming that the reader is familiar with both the primal and dual simplex methods, Zionts' crisscross method can easily be explained. ffl It can be initialized by any, possibly both primal and dual infeasible basis . If the basis is optimal, we are done. If the basis is not optimal , then there are some primal or dual infeasible variables. One might choose any of these. It is advised to choose once a primal and then a dual infeasible variable, if possible. ffl If the selected variable is dual infeasible, then it enters the basis and the leaving variable is chosen among the primal feasible variables in such a way that primal feasibility of the currently primal feasible variables is preserved. If no such basis exchange is possible another infeasible variable is selected. ffl If the selected variable is primal infeasible, then it leaves the basis and the entering variable is chosen among th
POSITIVELY ORIENTED MATROIDS ARE REALIZABLE
"... Dedicated to the memory of Michel Las Vergnas. Abstract. We prove da Silva’s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive mat ..."
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Dedicated to the memory of Michel Las Vergnas. Abstract. We prove da Silva’s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball. 1.
ACImplementation of the Reverse Search Vertex Enumeration Algorithm
, 1993
"... This report documents a C implementation of the reverse search vertex enumeration algorithm for convex polyhedra of Avis and Fukuda. The implementation uses multiple precision rational arithmetic and contains a few improvements over the original algorithm, resulting in a small reduction in the compl ..."
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This report documents a C implementation of the reverse search vertex enumeration algorithm for convex polyhedra of Avis and Fukuda. The implementation uses multiple precision rational arithmetic and contains a few improvements over the original algorithm, resulting in a small reduction in the complexity in most cases. For a polyhedron with n 0 inequalities in d nonnegative variables and nondegenerate origin, it finds all bases in time O(n 0d 2)per basis. It is also shown how the implementation can be improved to run in time O(n 0dmin(n 0, d)) time per basis for any polyhedron. This implementation can handle problems of quite large size, especially for simple polyhedra (where each basis corresponds to a vertex). Computational experience is included in the report. 1. Background Anew enumeration method called &quot;Reverse Search &quot; was recently introduced by Avis and Fukuda[1] (hereafter referred to as AF) and applied to the problem of finding all vertices of a
Positroids, noncrossing partitions, and positively oriented matroids
"... Abstract. We investigate the role that noncrossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a noncrossing partition on the ground set, and then freely placing the structure of a co ..."
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Abstract. We investigate the role that noncrossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a noncrossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, and we prove that the probability that a positroid on [n] is connected equals 1/e 2 asymptotically. We also prove da Silva’s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball. Résumé. Nous étudions le rôle des partitions sans croisements dans l’étude des positroïdes, une classe de matroïdes introduite par Postnikov. On montre que chaque positroïde peut être construit de manière unique par le choix d’une partition sans croisements de l’ensemble [n] ainsi que le choix d’un positroide connexe pour chacun des blocs de la partition. Nous utilisons ce résultat pour énumérer les positroïdes connexes, et nous prouvons que la probabilité qu’un positroïde sur [n] soit connexe est asymptotiquement égale à 1/e 2. Nous prouvons aussi une conjecture de 1987 dûe à da Silva: tout matroïde orienté positivement est un positroïde; autrement dit, il peut être réalisé par un ensemble de vecteurs dans un espace vectoriel réel. Il découle de ce résultat que la Grassmannienne matroïde positive (ou MacPhersonienne positive) est homéomorphe à un boule fermée.
Euclideaness and final polynomials in oriented matroid theory
 COMBINATORICA
, 1993
"... This paper deals with a geometric construction of algebraic nonrealizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (biquadratic) final polynomial [3], [5] for any noneuclidean oriented matroid. Here we apply the results of Edmonds, Fukuda a ..."
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This paper deals with a geometric construction of algebraic nonrealizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (biquadratic) final polynomial [3], [5] for any noneuclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning nondegenerate cycling of linear programms in noneuclidean oriented matroids.
Recollections on the discovery of the reverse search technique
"... Komei Fukuda and I discovered the idea for reverse search during conversations in Tokyo in October 1990. We were working on the vertex enumeration problem for convex polyhedra. At the time I was visiting Masao Iri at the University of Tokyo and Masakazu Kojima at Tokyo Institute of Technology, suppo ..."
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Komei Fukuda and I discovered the idea for reverse search during conversations in Tokyo in October 1990. We were working on the vertex enumeration problem for convex polyhedra. At the time I was visiting Masao Iri at the University of Tokyo and Masakazu Kojima at Tokyo Institute of Technology, supported by a JSPS/NSERC bilateral exchange. Komei was then working at the University of Tsukuba, Otsuka, which was a couple of subway stations from my office. So we met quite often. One day Komei visited me at my office in Todai and explained to me the crisscross method for solving linear programs, independently developed by S. Zionts [11], T. Terlaky [8, 9] and Z. Wang [10]. In this method one pivots in the hyperplane arrangement generated by the constraints of the linear program, without regard for feasibility thus differentiating it from the simplex method. Komei and Tomomi Matsui, a Ph.D. student at Tokyo Institute of Technology, had developed an elegant new proof of the convergence of the crisscross method [6], which Komei was explaining to me. Komei had drawn a linearrangement on the blackboard, along with the path the crisscross method would take from any given vertex to the optimum vertex of the LP. On the board all of these edges were shown in yellow with directions that eventually
LEASTINDEX ANTICYCLING RULES, LindAcR
, 1998
"... this paper. leastindex rules were designed for network flow problems, linear optimization problems, linear complementarity problems and oriented matroid programming problems. These classes will be considered in the sequel. ..."
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this paper. leastindex rules were designed for network flow problems, linear optimization problems, linear complementarity problems and oriented matroid programming problems. These classes will be considered in the sequel.