Integer linear programming

Integer linear programming refers to optimization problems of the form:

 minimize $f(\mathbf{x})$ with respect to $\mathbf{x}$ subject to $\mathbf{g(x) \leq 0}$ $\mathbf{h(x)=0}$ $\mathbf{x}\in\mathcal{Z}^n$

where $f(\mathbf{x})$, $\mathbf{g(x)}$ and $\mathbf{h(x)}$ are all affine functions of the vector $\mathbf{x}$, n is a positive integer, and $\mathcal{Z}$ is the set of integers.