How should we deal with strict inequalities in a linear programming problem? For example:
inequalities such as $ax< b$;
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6$\begingroup$ Add a tolerance, $\epsilon>0$ and try solving with $ax \leq b-\epsilon$. $\endgroup$– copper.hatJul 20, 2012 at 7:57
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$\begingroup$ @copper.hat Does tha apply to answer below? $\endgroup$– BCLCMar 3, 2016 at 3:40
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1$\begingroup$ @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.hatMar 3, 2016 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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3$\begingroup$ 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. $\endgroup$– equaegheJul 16, 2013 at 14:30
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