Abstract
So far, we have avoided using matrix notation to present linear programming problems and the simplex method. In this chapter, we shall recast everything into matrix notation. At the same time, we will emphasize the close relations between the primal and the dual problems.
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In this chapter, we have accomplished two tasks: (1) we have expressed the simplex method in matrix notation, and (2) we have reduced the information we carry from iteration to iteration to simply the list of basic variables together with current values of the primal basic variables and the dual nonbasic variables. In particular, it is not necessary to calculate explicitly all the entries of the matrix B -1 N.
What’s in a name? There are times when one thing has two names. So far in this book, we have discussed essentially only one algorithm: the simplex method (assuming, of course, that specific pivot rules have been settled on). But this one algorithm is sometimes referred to as the simplex method and at other times it is referred to as the revised simplex method. The distinction being made with this new name has nothing to do with the algorithm. Rather it refers to the specifics of an implementation. Indeed, an implementation of the simplex method that avoids explicit calculation of the matrix B — ’N is referred to as an implementation of the revised simplex method. We shall see in Chapter 8 why it is beneficial to avoid computing B-1N.
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© 2001 Robert J. Vanderbei
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Vanderbei, R.J. (2001). The Simplex Method in Matrix Notation. In: Linear Programming. International Series in Operations Research & Management Science, vol 37. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5662-3_6
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DOI: https://doi.org/10.1007/978-1-4757-5662-3_6
Publisher Name: Springer, Boston, MA
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