When is the polyhedron associated with a mixed-integer program different from the polyhedron associated with its LP-relaxation?

Question

I'm looking at the following definitions and observation:

Polyhedra associated with (MIP) and LP(MIP) are \begin{array}{c} P_{I P}(M I P):=\operatorname{conv}\left\{x \in \mathbb{R}^{n}: x \text { feasible for }(\mathrm{MIP})\right\} \\ P_{L P}(M I P):=\left\{x \in \mathbb{R}^{n}: x \text { feasible for } \mathrm{LP}(\mathrm{MIP})\right\} \end{array}

Observation:

But in the following example, shouldn't $P_{IP} = P_{LP}$? Shouldn't the convex hull of the solution set of the (MIP) include non-integral solutions, and hence is the same as $P_{LP}$? And if so, in the observation above, shouldn't the chain of relations be: "$...\geq ... = ... = ...$", instead of "$... = ... \geq ... = ...$"? How do I make sense of these discrepancies?

Answer 1

Consider the MIP with one variable $x$ and constraints $-\frac12\leq x\leq \frac12,\ x\in\mathbb{Z}$. Then $P_{IP}=\{0\}$ and $P_{LP}=[-\frac12,\frac12]$.

You always have $P_{IP}\subseteq P_{LP}$ but it is not always equality.