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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 ..."
Abstract

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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 algorithmbased 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.
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
Exterior point simplextype algorithms . . .
, 2015
"... Two decades of research led to the development of a number of efficient algorithms that can be classified as exterior point simplextype. This type of algorithms can cross over the infeasible region of the primal (dual) problem and find an optimal solution reducing the number of iterations needed. ..."
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Two decades of research led to the development of a number of efficient algorithms that can be classified as exterior point simplextype. This type of algorithms can cross over the infeasible region of the primal (dual) problem and find an optimal solution reducing the number of iterations needed. The main idea of exterior point simplextype algorithms is to compute two paths/flows. Primal (dual) exterior point simplextype algorithms compute one path/flow which is basic but not always primal (dual) feasible and the other is primal (dual) feasible but not always basic. The aim of this paper is to explain to the general OR audience, for the first time, the developments in exterior point simplextype algorithms for linear and network optimization problems, over the recent years. We also present other approaches that, in a similar way, do not preserve primal or dual feasibility at each iteration such as the monotonic buildup Simplex algorithms and the crisscross methods.