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I have been trying to find proof of the ratio test for Dual Simplex, that is, selecting the variable that enters the base.

Specifically, I was interested in knowing why by making that choice, reduced costs stay positive.

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    $\begingroup$ Have you looked at proofs in introductory LP textbooks? It is covered, for example, in section 4.5 of the second edition of Luenberger, "Linear and Nonlinear Programming". $\endgroup$
    – prubin
    Jan 16 at 21:39

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