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How should we deal with strict inequalities in a linear programming problem? For example:

inequalities such as $ax< b$;

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    $\begingroup$ Add a tolerance, $\epsilon>0$ and try solving with $ax \leq b-\epsilon$. $\endgroup$ – copper.hat Jul 20 '12 at 7:57
  • $\begingroup$ @copper.hat Does tha apply to answer below? $\endgroup$ – BCLC Mar 3 '16 at 3:40
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    $\begingroup$ @BCLC: In general, there will be no solution if the inequality is strict. So, what you do depends on what you want. The $\epsilon$ trick will work, but if the constraint is active, then the solution will not necessarily be optimal for the original problem. $\endgroup$ – copper.hat Mar 3 '16 at 4:07
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In general strict inequalities are not treated in linear programming problems, since the solution is not guaranteed to exist on corner points.

Consider the $1$-variable LPP: $Max$ $x$ subject to $x<3$. Now there does not exist any value of $x$ for which maximum is achieved and which lies in the feasible region.

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    $\begingroup$ 1. Linear programs are not necessarily about optimization, they can also be about feasibility. 2. Replacing ‘$\max$’ with ‘$\sup$’ evades the technical issue you point out. $\endgroup$ – equaeghe Jul 16 '13 at 14:30
  • $\begingroup$ $x \le 3 - \epsilon$ ? $\endgroup$ – BCLC Mar 3 '16 at 3:39

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