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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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    Add a tolerance, $\epsilon>0$ and try solving with $ax \leq b-\epsilon$.2012-07-20
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    @copper.hat Does tha apply to answer below?2016-03-03
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    @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.2016-03-03

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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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    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.2013-07-16
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    $x \le 3 - \epsilon$ ?2016-03-03