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I started to work a bit with linear programming methods and would like to know why we can't use strict inequalities in our constraints, i.e., why is the equivalence in constraints excluded?

What can I do if I need a strict inequality in my constraints to formulate a problem? Can strict inequalities reformulated as simple inequalities?

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If the domain is not closed such as in the case of strict inequalities, the optimal value need not exists.

For example consider $\min x$ subject to $x>0$.

We can't find the smallest value as $x$ can get arbitrarily small and positive.

We can pick a small positive quantity and solve $\min x$ subject to $x \ge \epsilon$ instead if you desire a positive quantity but it is no longer the same problem.

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  • $\begingroup$ Does that mean, that I can reformulate the following constraints $x_1 > x_2$, $x_1 > x_3$ into $x_1 \geq x_2 + \epsilon$ and $x_1 \geq x_3 + \epsilon$? The point is that I really don't want $x_1 = x_2 = x_3$ but which could happen using the approach above. How could I avoid equality here? $\endgroup$ – Samuel Apr 12 at 9:46
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    $\begingroup$ yup, it is an approximation to the infimum. A smaller $\epsilon$ is preferred. $\endgroup$ – Siong Thye Goh Apr 12 at 9:49

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