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.

翻译：$L_{\infty}$星形偏差是一种被广泛研究的度量，用于量化点集分布的均匀性。在偏差研究领域，构造该度量的最优点集被视为极具挑战的问题。实际上，目前仅在二维空间中$n\leq 6$以及更高维空间中$n \leq 2$的情况下已知最优解。本文提出基于数学规划的框架，用于构造尽可能低$L_{\infty}$星形偏差的点集。首先，我们给出两种构造最优点集的模型，并证明始终存在满足"任意两点无坐标重合"性质的最优解。进一步，我们将模型推广至其他度量，如极值偏差与周期性偏差。针对$L_{\infty}$星形偏差，我们在二维空间中计算了最多21个点的最优解，在三维空间中则实现了最多8个点的最优解。对于$d=2$且$n\ge 7$的情况，这些点集的偏差比当前最优解降低约50%，并展现出完全不同的结构特征。