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Consider a standard LP minimization problem of the form

$$\begin{array}{ll} \text{minimize} & c^\top x\\ \text{subject to} & A x = b\\ & x \geq 0\end{array}$$

Should I expect, on average, an improvement in the number of iterations needed to achieve optimality if adding a (tight enough) constraint of the form $c^\top x \leq m + \varepsilon$ to it? By "tight enough" I mean that $m$ is the optimal value of the problem and $\varepsilon$ is small in some sense.

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