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

inequalities such as $ax< b$;

  • 6
    $\begingroup$ 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
  • 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.hat Mar 3 '16 at 4:07

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.

  • 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. $\endgroup$ – equaeghe Jul 16 '13 at 14:30
  • $\begingroup$ $x \le 3 - \epsilon$ ? $\endgroup$ – BCLC Mar 3 '16 at 3:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.