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11
CrissCross Methods: A Fresh View on Pivot Algorithms
 Mathematical Programming
, 1997
"... this paper is to present mathematical ideas and ..."
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 9 (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 algorithm for o ..."
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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.
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 3 (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
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 sign ..."
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Cited by 1 (1 self)
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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...
The Finite CrissCross Method for Hyperbolic Programming
 Informatica, Technische Universiteit Delft, The Netherlands
, 1996
"... In this paper the finite crisscross method is generalized to solve hyperbolic programming problems. Just as in the case of linear or quadratic programming the crisscross method can be initialized with any, not necessarily feasible basic solution. Finiteness of the procedure is proved under the ..."
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In this paper the finite crisscross method is generalized to solve hyperbolic programming problems. Just as in the case of linear or quadratic programming the crisscross method can be initialized with any, not necessarily feasible basic solution. Finiteness of the procedure is proved under the usual mild assumptions. Some small numerical examples illustrate the main features of the algorithm. Key words: hyperbolic programming, pivoting, crisscross method iii 1 Introduction The hyperbolic (fractional linear) programming problem is a natural generalization of the linear programming problem. The linear constraints are kept, but the linear objective function is replaced by a quotient of two linear functions. Such fractional linear objective functions arise in economical models when the goal is to optimize profit/allocation type functions (see for instance [12]). The objective function of the hyperbolic programming problem is neither linear nor convex, however there are several ...
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 , then 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
Principal Pivoting Methods For Linear Complementarity Problems, PCPLCP
"... timization problem min ae c T x + 1 2 x T Qx : Ax b; x 0 oe ; where Q is a positive semidefinite, symmetric matrix, then M = ` 0 A \GammaA T Q ' and q = ` \Gammab c ' : Here M is a positive semidefinite bisymmetric matrix. Bisymmetry means that the matrix has a block diago ..."
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timization problem min ae c T x + 1 2 x T Qx : Ax b; x 0 oe ; where Q is a positive semidefinite, symmetric matrix, then M = ` 0 A \GammaA T Q ' and q = ` \Gammab c ' : Here M is a positive semidefinite bisymmetric matrix. Bisymmetry means that the matrix has a block diagonal structure, and it is the sum of a symmetric block diagonal positive semidefinite, and a skew symmetric block diagonal matrix. Some other classes of solvable LCPs are problems, when M is a ffl P matrix ; ffl sufficient matrix or, equivalently, a P
K. Fukuda and T. Terlaky ISSN 09225641 Reports of the Faculty of Technical Mathematics and Informatics 99?? Delft January, 1999
"... this paper the nondegeneracy assumption is removed. Our constructive proof relies on similar ideas that were developed for strongly polynomial basis identification techniques in interior point methods [10,11]. 3 In the rest of this section we fix our notations and give formal definitions. In Sectio ..."
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this paper the nondegeneracy assumption is removed. Our constructive proof relies on similar ideas that were developed for strongly polynomial basis identification techniques in interior point methods [10,11]. 3 In the rest of this section we fix our notations and give formal definitions. In Section 2.1 for the primal and in Section 2.2 for the dual feasibility problem it will be shown that from any basis to a feasible basis an admissible pivot sequence exists whose length is bounded by n and m, respectively. Our main result, in Section 3 shows that the answer to Question 1 is positive, there is always an admissible pivot sequence consisting of not more then m + n
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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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.