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18
A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra
, 1990
"... 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 223 (33 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) 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(n²dv) 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 14 (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.
The Theory of Linear Programming: Skew Symmetric SelfDual Problems and the Central Path
, 1994
"... The literature in the field of interior point methods for linear programming has been almost exclusively algorithm oriented. Recently Guler, Roos, Terlaky and Vial presented a complete duality theory for linear programming based on the interior point approach. In this paper we present a more simple ..."
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Cited by 12 (6 self)
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The literature in the field of interior point methods for linear programming has been almost exclusively algorithm oriented. Recently Guler, Roos, Terlaky and Vial presented a complete duality theory for linear programming based on the interior point approach. In this paper we present a more simple approach which is based on an embedding of the primal problem and its dual into a skew symmetric selfdual problem. This embedding is essentially due Ye, Todd and Mizuno. First we consider a skew symmetric selfdual linear program. We show that the strong duality theorem trivially holds in this case. Then, using the logarithmic barrier problem and the central path, the existence of a strictly complementary optimal solution is proved. Using the embedding just described, we easily obtain the strong duality theorem and the existence of strictly complementary optimal solutions for general linear programming problems. Key words. Linear programming, interior points, skew symmetric matrix, self...
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 11 (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 9 (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 7 (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
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 6 (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.
Developments in Linear and Integer Programming
"... In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimization models to aid decision making and over this period the size of problems that can be solved has increased dramatically, ..."
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Cited by 6 (0 self)
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In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimization models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimization systems embedded in other decision support tools has made online decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimization and other technologies.
On the Existence of a Short Admissible Pivot Sequences for Feasibility and Linear Optimization Problems
, 1999
"... this paper, for the feasibility problem, we prove the existence of a short admissible pivot sequence from an arbitrary basis to a feasible basis. Regarding the general LP problem, the existence of a short admissible pivot sequence from an arbitrary basis to an optimal basis is proved without any non ..."
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Cited by 4 (1 self)
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this paper, for the feasibility problem, we prove the existence of a short admissible pivot sequence from an arbitrary basis to a feasible basis. Regarding the general LP problem, the existence of a short admissible pivot sequence from an arbitrary basis to an optimal basis is proved without any nondegeneracy assumptions. Our constructive proofs are based on techniques that are used in stronglypolynomial basis identification schemes of interior point methods. The result can be regarded as an admissible pivot version of the dstep
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 3 (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.