Building upon the exact methods presented in our earlier work [J. Complexity, 2022], we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While we prove that the heuristic does not necessarily return an optimal solution, we obtain very promising results for all tested dimensions. For example, for moderate point set sizes $30 \leq n \leq 240$ in dimension 6, we obtain point sets with $L_{\infty}$ star discrepancy up to 35% better than that of the first $n$ points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We also provide a comparison with a recent energy functional introduced by Steinerberger [J. Complexity, 2019], showing that our heuristic performs better on all tested instances.

翻译：基于我们早期工作中提出的精确方法[J. Complexity，2022]，我们针对星形差异子集选择问题引入了一种启发式方法。该启发式方法通过每次替换当前最优子集中的一个元素来逐步改进。虽然我们证明该方法不一定能返回最优解，但所有测试维度均获得了非常理想的结果。例如，在维度6中，对于中等规模点集$30 \leq n \leq 240$，我们获得的点集$L_{\infty}$星形差异比Sobol'序列前$n$个点构成的点集最多可降低35%。该方法适用于所有维度，其主要限制在于差异计算算法的精度。我们还与Steinerberger [J. Complexity，2019]近期提出的能量泛函进行了比较，结果表明我们的启发式方法在所有测试实例上均表现更优。