Papers › Constructing Optimal L_∞ Star Discrepancy Sets
Constructing Optimal L_∞ Star Discrepancy Sets
François Clément, Carola Doerr, Kathrin Klamroth, Luís Paquete
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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.
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