# Using strict inequalities for constraints in linear programming

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?

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.
• 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? – Samuel Apr 12 at 9:46
• yup, it is an approximation to the infimum. A smaller $\epsilon$ is preferred. – Siong Thye Goh Apr 12 at 9:49