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17
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...
On the Finiteness of the CrissCross Method
 European Journal of Operations Research
, 1989
"... . In this short paper, we prove the finiteness of the crisscross method by showing a certain binary number of bounded digits associated with each iteration increases monotonically. This new proof immediately suggests the possibility of relaxing the pivoting selection in the crisscross method witho ..."
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Cited by 7 (2 self)
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. In this short paper, we prove the finiteness of the crisscross method by showing a certain binary number of bounded digits associated with each iteration increases monotonically. This new proof immediately suggests the possibility of relaxing the pivoting selection in the crisscross method without sacrificing the finiteness. Key Words: linear programming. simplex method, finite pivoting rules. 1 The CrissCross Method Let A be an m2 n matrix. Let E be the index set of columns of the matrix A; and f; g be two distinct members of E: Here we consider the standard form linear program: (P ) maximize x f (1.1) subject to A x = 0; (1.2) x g = 1; (1.3) x j 0; 8 j 2 E 0 ff; gg: (1.4) A vector x is said to be feasible if it satisfies the constraints (1.2), (1.3), and (1.4). If a linear program has a feasible solution, then it is called feasible, otherwise it is called infeasible. For any linear program, we will refer to following three situations as characters: 3 Supported by Grant...
The Role of Pivoting in Proving Some Fundamental Theorems of Linear Algebra
 Linear Algebra and Its Applications 151
, 1991
"... This paper contains a new approach to some classical theorems of linear algebra (Steinitz, matrix rank, RoucheKroneckerCapelli, Farkas, Weyl, Minkowski). The constructive proofs are based on pivoting. Defining pivoting in a more general way  using generating tableaux  made it possible to give a ..."
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Cited by 6 (1 self)
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This paper contains a new approach to some classical theorems of linear algebra (Steinitz, matrix rank, RoucheKroneckerCapelli, Farkas, Weyl, Minkowski). The constructive proofs are based on pivoting. Defining pivoting in a more general way  using generating tableaux  made it possible to give a new proof for Steinitz theorem as well. Our pivot selection strategies are based essentially on Bland's [2] minimal index rule. The famous theorems of Farkas, Weyl and Minkowski are proved by using pivot tableaux. Theorem 4.1 is essentially a new, very simple form of the alternative theorem of linear inequalities, and its proof is a pretty application of the minimal index rule. One can apply this theorem and its proof to combinatorial structures (for example to oriented matroids) as well (KlafszkyTerlaky [9]). The presented algorithms are mainly not efficient computationally (see e.g. Roos [13] for an exponential example), but they are surpisingly simple. We will use the symbols 0; +; \Gamma; \Phi; \Psi introduced by BalinskiTucker [1], which denote zero, positive, negative, nonnegative and nonpositive numbers respectively. On the other hand Gale's [7] notations will be used, so matrices and vectors are denoted by capital and small Latin letters and their components are denoted by the corresponding Greek letters. Index sets are denoted by I and J (with proper subscripts) and the cardinality of an index set J is denoted by k J k. 2 Pivoting
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 Monotonic BuildUp Simplex Algorithm for Linear Programming
, 1991
"... We devise a new simplex pivot rule which has interesting theoretical properties. Beginning with a basic feasible solution, and any nonbasic variable having a negative reduced cost, the pivot rule produces a sequence of pivots such that ultimately the originally chosen nonbasic variable enters the ba ..."
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Cited by 5 (1 self)
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We devise a new simplex pivot rule which has interesting theoretical properties. Beginning with a basic feasible solution, and any nonbasic variable having a negative reduced cost, the pivot rule 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. The pivot rule thus monotonically builds up to a dual feasible, and hence optimal, basis. A surprising property of the pivot rule is that the pivot sequence results in intermediate bases which are neither primal nor dual feasible. We prove correctness of the procedure, give a geometric interpretation, and relate it to other pivoting rules for linear programming.
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...
New Variants Of Finite CrissCross Pivot Algorithms For Linear Programming
, 1997
"... In this paper we generalize the socalled firstinlastout pivot rule and the mostoftenselectedvariable pivot rule for the simplex method, as proposed in Zhang [13], to the crisscross pivot setting where neither the primal nor the dual feasibility is preserved. The finiteness of the new crisscr ..."
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Cited by 2 (0 self)
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In this paper we generalize the socalled firstinlastout pivot rule and the mostoftenselectedvariable pivot rule for the simplex method, as proposed in Zhang [13], to the crisscross pivot setting where neither the primal nor the dual feasibility is preserved. The finiteness of the new crisscross pivot variants is proven.