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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
Combinatorial Maximum Improvement Algorithm for LP and LCP
, 1995
"... this paper, we show how one can design new pivot algorithms for solving the LP and the LCP. In particular, we are interested in combinatorial pivot algorithms which solve the LP and a certain class of LCP's. Here, a pivot algorithm is called combinatorial if the pivot choice depends only on the ..."
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this paper, we show how one can design new pivot algorithms for solving the LP and the LCP. In particular, we are interested in combinatorial pivot algorithms which solve the LP and a certain class of LCP's. Here, a pivot algorithm is called combinatorial if the pivot choice depends only on the signs of entries of their dictionaries. The best source of combinatorial pivot algorithms is in the theory of oriented matroid (OM) programming [Bla77a, Edm94, Fuk82, FT92, LL86, Ter87, Tod85, Wan87]. The wellknown Bland's pivot rule [Bla77b] for the simplex method can be considered as a combinatorial algorithm, but it is not a typical one. The main characteristic of the "OM" algorithms is that the feasibility may not be preserved at all in both primal and dual problem, and the finiteness of the algorithms is guaranteed by some purely combinatorial improvement argument rather than by the reasoning based on the increment of the objective function value. One immediate advantage of combinatorial algorithms is that the degeneracy does not have to be treated separately. Thus a very simple combinatorial algorithm, such as the crisscross method [Ter87, Wan87], solves the general LP correctly and yields one of the simplest proofs of the strong duality theorem. There is a wellnoted disadvantage of combinatorial algorithms. The number of pivot operations to solve the LP tends to grow rapidly in practice. Furthermore it is often quite easy to construct a class of LP's for which a given combinatorial algorithm takes an exponential number of pivot operations in the input size. In this paper, we review the finiteness proof of combinatorial algorithms and study a new algorithm in the class. The key ingredients of the new algorithm are "history dependency" and "largest combinatorial improveme...
Towards a Unified Framework for Randomized Pivoting Algorithms in Linear Programming
 IN OPERATIONS RESEARCH PROCEEDINGS
, 1998
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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.
POSITIVELY ORIENTED MATROIDS ARE REALIZABLE
, 2013
"... 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 homeomor ..."
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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.
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
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