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?
If the domain is not closed such as in the case of strict inequalities, the optimal value need not exists.
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.