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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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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 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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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 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.
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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. 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
The CrissCross Method Can Take \Omega (nd) Pivots
, 2004
"... Abstract Using deformed products of arrangements, we construct a family of linear programs with ninequalities in! d on which, in the worstcase, the leastindex crisscross method requires \Omega (nd) (for fixed d) pivots to reach optimality. ..."
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Abstract Using deformed products of arrangements, we construct a family of linear programs with ninequalities in! d on which, in the worstcase, the leastindex crisscross method requires \Omega (nd) (for fixed d) pivots to reach optimality.
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.