Results 1  10
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15
A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra
, 1992
"... We present a new piv otbased algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following prope ..."
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Cited by 193 (29 self)
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We present a new piv otbased algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties: (a) Virtually no additional storage is required beyond the input data; (b) The output list produced is free of duplicates; (c) The algorithm is extremely simple, requires no data structures, and handles all degenerate cases; (d) The running time is output sensitive for nondegenerate inputs; (e) The algorithm is easy to efficiently parallelize. For example, the algorithm finds the v vertices of a polyhedron in R d defined by a nondegenerate system of n inequalities (or dually, the v facets of the convex hull of n points in R d,where each facet contains exactly d given points) in time O(ndv) and O(nd) space. The v vertices in a simple arrangement of n hyperplanes in R d can be found in O(n 2 dv) time and O(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.
Some Generalizations Of The CrissCross Method For Quadratic Programming
 MATH. OPER. UND STAT. SER. OPTIMIZATION
, 1992
"... Three generalizations of the crisscross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. ..."
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Cited by 13 (8 self)
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Three generalizations of the crisscross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite crisscross method, based on leastindex resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used. In the last section some special cases and two further variants of the quadratic crisscross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty's `Bard type schema' in the P matrix case.
Linear Complementarity and Oriented Matroids
 Journal of the Operational Research Society of Japan
, 1990
"... A combinatorial abstraction of the linear complementarity theory in the setting of oriented matroids was rst considered by M.J. Todd. In this paper, we take a fresh look at this abstraction, and attempt to give a simple treatment of the combinatorial theory of linear complementarity. We obtain new t ..."
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Cited by 12 (8 self)
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A combinatorial abstraction of the linear complementarity theory in the setting of oriented matroids was rst considered by M.J. Todd. In this paper, we take a fresh look at this abstraction, and attempt to give a simple treatment of the combinatorial theory of linear complementarity. We obtain new theorems, proofs and algorithms in oriented matroids whose specializations to the linear case are also new. For this, the notion of suciency of square matrices, introduced by Cottle, Pang and Venkateswaran, is extended to oriented matroids. Then, we prove a sort of duality theorem for oriented matroids, which roughly states: exactly one of the primal and the dual system has a complementary solution if the associated oriented matroid satisfies "weak" sufficiency. We give two different proofs for this theorem, an elementary inductive proof and an algorithmic proof using the crisscross method which solves one of the primal or dual problem by using surprisingly simple pivot rules (without any pertur...
A Survey on Pivot Rules for Linear Programming
 ANNALS OF OPERATIONS RESEARCH. (SUBMITTED
, 1991
"... The purpose of this paper is to survey the various pivot rules of the simplex method or 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 the finiteness property of simplex type pivot rules. Th ..."
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Cited by 10 (1 self)
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The purpose of this paper is to survey the various pivot rules of the simplex method or 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 the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...
The Linear Complementarity Problem, Sufficient Matrices and the CrissCross Method
, 1990
"... Specially structured Linear Complementarity Problems (LCP's) and their solution by the crisscross method are examined in this paper. The crisscross method is known to be finite for LCP's with positive semidefinite bisymmetric matrices and with Pmatrices. It is also a simple finite algor ..."
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Cited by 6 (4 self)
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Specially structured Linear Complementarity Problems (LCP's) and their solution by the crisscross method are examined in this paper. The crisscross method is known to be finite for LCP's with positive semidefinite bisymmetric matrices and with Pmatrices. It is also a simple finite algorithm for oriented matroid programming problems. Recently Cottle, Pang and Venkateswaran identified the class of (column, row) sufficient matrices. They showed that sufficient matrices are a common generalization of P and PSDmatrices. Cottle also showed that the principal pivoting method (with a clever modification) can be applied to row sufficient LCP's. In this paper the finiteness of the crisscross method for sufficient LCP's is proved. Further it is shown that a matrix is sufficient if and only if the crisscross method processes all the LCP's defined by this matrix and all the LCP's defined by the transpose of this matrix and any parameter vector.
A Polynomial Method of Weighted Centers for Convex Quadratic Programming
 JOURNAL OF INFORMATION & OPTIMIZATION SCIENCES
, 1991
"... A generalization of the weighted central pathfollowing method for convex quadratic programming is presented. This is done by uniting and modifying the main ideas of the weighted central pathfollowing method for linear programming and the interior point methods for convex quadratic programming. B ..."
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Cited by 3 (2 self)
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A generalization of the weighted central pathfollowing method for convex quadratic programming is presented. This is done by uniting and modifying the main ideas of the weighted central pathfollowing method for linear programming and the interior point methods for convex quadratic programming. By means of the linear approximation of the weighted logarithmic barrier function and weighted inscribed ellipsoids, `weighted' trajectories are defined. Each strictly feasible primal dual point pair define such a weighted trajectory. The algorithm can start in any strictly feasible primaldual point pair that defines a weighted trajectory, which is followed through the algorithm. This algorithm has the nice feature, that it is not necessary to start the algorithm close to the central path and so additional transformations are not needed. In return, the theoretical complexity of our algorithm is dependent on the position of the starting point. Polynomiality is proved under the usual mild conditions.
Basis and Tripartition Identification for Quadratic Programming and Linear Complementarity Problems  From an interior solution to an optimal basis and viceversa
, 1996
"... Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximal complementary solutions. Maximal complementary solutions can be characterized by optimal (tri)partitions. On the other hand, the solutions provided by simplexb ..."
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Cited by 3 (2 self)
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Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximal complementary solutions. Maximal complementary solutions can be characterized by optimal (tri)partitions. On the other hand, the solutions provided by simplexbased pivot algorithms are given in terms of complementary bases. A basis identification algorithm is an algorithm which generates a complementary basis, starting from any complementary solution. A tripartition identification algorithm is an algorithm which generates a maximal complementary solution (and its corresponding tripartition), starting from any complementary solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal...
Parametric linear programming and anticycling pivoting rules
 MATHEMATICAL PROGRAMMING
, 1988
"... The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore, restrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the choic ..."
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Cited by 2 (0 self)
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The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore, restrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the choice of exiting variables. Using ideas from parametric linear programming, we develop anticycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule. That is, any variable that ties for the ratio rule can leave the basis. A similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables.
On Circuit Valuation of Matroids
, 2000
"... The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R[f01g)valued vectors defined on the circuits of the un ..."
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Cited by 1 (1 self)
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The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R[f01g)valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of (R [ f01g) valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.
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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Cited by 1 (0 self)
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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