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：$L_{\infty}$星偏差是用于量化点集分布均匀性的一个被广泛研究的度量。在偏差研究领域，构造该度量的最优点集被视为一个极具挑战性的问题。事实上，截至目前，已知的最优点集仅存在于二维情形下$n \leq 6$的点集以及更高维度下$n \leq 2$的点集。本文引入数学规划公式，旨在构造具有尽可能低的$L_{\infty}$星偏差的点集。首先，我们提出两种构造最优集合的模型，并证明总存在满足任意两点坐标均不重合性质的最优集合。随后，我们展示了模型对其他度量（如极值偏差和周期偏差）的推广可能性。针对$L_{\infty}$星偏差，我们成功计算了二维情形下多达21个点以及三维情形下多达8个点的最优点集。对于$d=2$且$n \geq 7$的点集，这些点集的偏差比当前最佳点集低约50%，并展现出截然不同的结构特征。