Dropping Upper Bounds in a Linear Program

Question

Can anyone explain why usually, in a Linear Program, the upperbound constraints are "redundant" and then they can be dropped?

For example, consider:

http://en.wikipedia.org/wiki/Set_cover_problem#Integer_linear_program_formulation

once I take the LP relaxation and then $x_{S} \in [0,1]$ (instead of taking either value $1$ or $0$, $x$ will take value from that continous interval), we have that the upper bound is called redundant and can be dropped and so it is enough to set $x_{S} \geq 0$.

Why does this hold intuitively?

Answer 1

For this particular problem, the objective function is to minimize the sum of the $x_S$. If you took a feasible solution with any $x_S>1$, then the objective value would be strictly better if you set $x_S=1$, and the solution would still be feasible. So in this case the upper bound constraints $x_S\leqslant 1$ are redundant.