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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 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.
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.
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
Finite Pivot Algorithms and Feasibility
, 2001
"... This thesis studies the classical finite pivot methods for solving linear programs and their efficiency in attaining primal feasibility. We review Dantzig’s largestcoefficient simplex method, Bland’s smallestindex rule, and the leastindex crisscross method. We present the b'rule: a simple ..."
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This thesis studies the classical finite pivot methods for solving linear programs and their efficiency in attaining primal feasibility. We review Dantzig’s largestcoefficient simplex method, Bland’s smallestindex rule, and the leastindex crisscross method. We present the b'rule: a simple algorithm based on Bland’s smallest index rule for solving systems of linear inequalities (feasibility of linear programs). We prove that the b'rule is finite, from which we then prove Farka’s Lemma, the Duality Theorem for Linear Programming, and the Fundamental Theorem of Linear Inequalities. We present experimental results that compare the speed of the b'rule to the classical methods. i Resumé Cette thèse étudie l’éfficacité des méthodes classiques finies des pivots qui résout les problèmes de programmation linéaire pour atteindre une solution admissible. Nous passons en revue la méthode du simplèxe du plusgrandcoefficient de Dantzig, la règle
LEASTINDEX ANTICYCLING RULES, LindAcR
, 1998
"... this paper. leastindex rules were designed for network flow problems, linear optimization problems, linear complementarity problems and oriented matroid programming problems. These classes will be considered in the sequel. ..."
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this paper. leastindex rules were designed for network flow problems, linear optimization problems, linear complementarity problems and oriented matroid programming problems. These classes will be considered in the sequel.
1 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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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.