Constructing Optimal L∞ Star Discrepancy Sets

The L∞ 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≤ 6 in dimension 2 and n ≤ 2 for higher dimensions. We introduce in this paper mathematical programming formulations to construct point sets with as low L∞ 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∞ 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≥ 7 points, these point sets have around a 50% lower discrepancy than the current best point sets, and show a very different structure.