  The $L_{\infty}$ star discrepancy is a very well-studied measure used to
quantify the uniformity of a point set distribution. Constructing optimal point
sets for this measure is seen as a very hard problem in the discrepancy
community. Indeed, optimal point sets are, up to now, known only for $n\leq 6$
in dimension 2 and $n \leq 2$ for higher dimensions. We introduce in this paper
mathematical programming formulations to construct point sets with as low
$L_{\infty}$ star discrepancy as possible. Firstly, we present two models to
construct optimal sets and show that there always exist optimal sets with the
property that no two points share a coordinate. Then, we provide possible
extensions of our models to other measures, such as the extreme and periodic
discrepancies. For the $L_{\infty}$ star discrepancy, we are able to compute
optimal point sets for up to 21 points in dimension 2 and for up to 8 points in
dimension 3. For $d=2$ and $n\ge 7$ points, these point sets have around a 50%
lower discrepancy than the current best point sets, and show a very different
structure.
