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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...
A Column Generation Approach to Bus Driver Scheduling
, 1996
"... This paper outlines an alternative solution method which has been incorporated into a system which originated from IMPACS. Improved results on a selection of real bus driver problems are presented. THE DRIVER SCHEDULING PROBLEM ..."
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This paper outlines an alternative solution method which has been incorporated into a system which originated from IMPACS. Improved results on a selection of real bus driver problems are presented. THE DRIVER SCHEDULING PROBLEM
An Efficient SteepestEdge Simplex Algorithm for SIMD Computers
"... This paper proposes a new parallelization of the Primal and Dual Simplex algorithms for Linear Programming (LP) problems on massively parallel SingleInstruction MultipleData (SIMD) computers. The algorithms are based on the SteepestEdge pivot selection method and the tableau representation of ..."
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This paper proposes a new parallelization of the Primal and Dual Simplex algorithms for Linear Programming (LP) problems on massively parallel SingleInstruction MultipleData (SIMD) computers. The algorithms are based on the SteepestEdge pivot selection method and the tableau representation of the constraint matrix. The initial canonical tableau is formed on an attached scalar host unit, and then partitioned into a rectangular grid of submatrices and distributed to the individual Processor Element (PE) memories. In the beginning of the parallel algorithm key portions of the simplex tableau are partially replicated and stored along with the submatrices on each one of the PEs. The SIMD simplex algorithm iteratively selects a pivot element and carriesout a simplex computation step until the optimal solution is found, or when unboundedness of the LP is established. The SteepestEdge pivot selection technique utilizes information mainly from local replicas to search for the next pivot element. The pivot row and column are selectively broadcasted to the PEs before a pivot computation step, by efficiently utilizing the geometry of the toroidal mesh interconnection network. Every individual PE maintains locally and keeps consistent its replicas so that interprocessor communication due to data dependencies is further reduced. The presence of a pipelined inteconnection network, like the mesh network of MP1 and MP2 MasPar models allows the global reduction operations necessary in the selection of pivot columns and rows to be performed in time O(log nR + log nC ), in (nR \Theta nC ) PE arrays. This particular combination of pivot selection, matrix representation, and selective data replication is shown to be highly efficient in the solution of linear prog...
A Piecewise Linear Dual Phase1 Algorithm for the Simplex Method with All Types of Variables
"... A dual phase1 algorithm for the simplex method that handles all types of variables is presented. In each iteration it maximizes a piecewise linear function of dual infeasibilities in order to make the largest possible step towards dual feasibility with a selected outgoing variable. The new method c ..."
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A dual phase1 algorithm for the simplex method that handles all types of variables is presented. In each iteration it maximizes a piecewise linear function of dual infeasibilities in order to make the largest possible step towards dual feasibility with a selected outgoing variable. The new method can be viewed as a generalization of traditional phase1 procedures. It is based on the multiple use of the expensively computed pivot row. By small amount of extra work per iteration, the progress it can make is equivalent to many iterations of the traditional method. In addition to this main achievement it has some further important and favorable features, namely, it is very efficient in coping with degeneracy and numerical difficulties. Both theoretical and computational issues are addressed. Examples are also given that demonstrate the power and flexibility of the method.
A General Pricing Scheme for the Simplex Method
, 2001
"... Pricing is a term in the simplex method for linear programming used to refer to the step of checking the reduced costs of nonbasic variables. If they are all of the `right sign' the current basis (and solution) is optimal, if not, this procedure selects a candidate vector that looks profitable ..."
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Pricing is a term in the simplex method for linear programming used to refer to the step of checking the reduced costs of nonbasic variables. If they are all of the `right sign' the current basis (and solution) is optimal, if not, this procedure selects a candidate vector that looks profitable for inclusion in the basis. While theoretically the choice of any profitable vector will lead to a finite termination (provided degeneracy is handled properly) but the number of iterations until termination depends very heavily on the actual choice (which is defined by the selection rule applied). Pricing has long been an area of heuristics to help make better selection. As a result, many different and sophisticated pricing strategies have been developed, implemented and tested. So far none of them is known to be dominating all others in all cases. Therefore, advanced simplex solvers need to be equipped with many strategies so that the most suitable one can be activated for each individual problem instance. In this paper we present a general pricing scheme. It creates a large flexibility in pricing. It is controlled by three parameters. With different settings of the parameters many of the known strategies can be reproduced as special cases. At the same time, the framework makes it possible to define new strategies or variants of them. The scheme is equally applicable to general and network simplex algorithms.
A Penalty Based Simplex Method for Linear Programming
, 1995
"... We give a general description of a new advanced implementation of the simplex method for linear programming. The method "decouples" a notion of the simplex basic solution into two independent entities: a solution and a basis . This generalization makes it possible to incorporate new strate ..."
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We give a general description of a new advanced implementation of the simplex method for linear programming. The method "decouples" a notion of the simplex basic solution into two independent entities: a solution and a basis . This generalization makes it possible to incorporate new strategies into the algorithm since the iterates no longer need to be the vertices of the simplex. An advantage of such approach is a possibility of taking steps along directions that are not simplex edges (in principle they can even cross the interior of the feasible set). It is exploited in our new approach to finding the initial solution in which global infeasibility is handled through a dynamically adjusted penalty term. We present several new techniques that have been incorporated into the method. These features include: ffl previously mentioned method for finding an initial solution, ffl an original approximate steepest edge pricing algorithm, ffl dynamic adjustment of the penalty term. The presenc...