LP relaxation for ILP\IP (integer linear programming)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a lower bound for the ILP. Why this is correct?

I do understand that a feasible solution for the ILP is a feasible solution to the LP, and the reversed is not always so, i.e. a feasible solution for the LP is not necessarily a feasible solution for the ILP.

Can one please point out and explain briefly why it is so?

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As you say, a feasible solution for the ILP is a feasible solution for the LP. So if the LP has an optimal solution with objective value $\alpha$, this implies there is no feasible solution for the LP with objective value $< \alpha$, and in particular no feasible solution for the ILP with objective value $<\alpha$. That's what it means for $\alpha$ to be a lower bound for the ILP.
For example, take this simple ILP: minimize $2 x$ subject to $2 x \ge 3$, $x$ an integer. Leaving out the "integer" requirement, the linear programming problem has optimal solution $x=3/2$ with objective value $3$. That says there is no real solution with $2x < 3$, and therefore no integer solution with $2x < 3$, so $3$ is a lower bound on the objective values for the ILP. Of course the actual optimal solution for the ILP has objective value $4$ in this case. –  Robert Israel Jul 29 '12 at 22:56