Abstract
A pivotal algebra algorithm is given and finite convergence shown for finding a vector which satisfies an arbitrary system of linear equations and/or inequalities. A modified form of the algorithm, obtained by introducing a redundant equation, is then shown to be a way to describe phase I of the simplex method without reference to artificial variables or an artificial objective function.
The hypothesis is introduced that in each pivot stage each row of the tableau has equal probability of being chosen as the pivot row. Under this assumption the expected value of the ratio of the number of pivot stages to the number of rows should grow with the natural Log of the number of rows.
Use of the algorithm in proving theorems of the alternative is indicated.
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References
G.B. Dantzig,Linear programming and extensions (Princeton University Press, 1963), pp. 136–140.
O.L. Mangasarian,Nonlinear programming (McGraw-Hill, New York, 1969) pp. 27–37.
P. Wolfe and L. Cutler, “Experiments in linear programming,” in:Recent advances in mathematical programming, Eds. R.L. Groves and P. Wolfe (McGraw-Hill, New York, 1963) pp. 177–200.
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Orden, A. On the solution of linear equation/inequality systems. Mathematical Programming 1, 137–152 (1971). https://doi.org/10.1007/BF01584083
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DOI: https://doi.org/10.1007/BF01584083