The $L_{\infty}$ star discrepancy is a measure for the regularity of a finite set of points taken from $[0,1)^d$. Low discrepancy point sets are highly relevant for Quasi-Monte Carlo methods in numerical integration and several other applications. Unfortunately, computing the $L_{\infty}$ star discrepancy of a given point set is known to be a hard problem, with the best exact algorithms falling short for even moderate dimensions around 8. However, despite the difficulty of finding the global maximum that defines the $L_{\infty}$ star discrepancy of the set, local evaluations at selected points are inexpensive. This makes the problem tractable by black-box optimization approaches. In this work we compare 8 popular numerical black-box optimization algorithms on the $L_{\infty}$ star discrepancy computation problem, using a wide set of instances in dimensions 2 to 15. We show that all used optimizers perform very badly on a large majority of the instances and that in many cases random search outperforms even the more sophisticated solvers. We suspect that state-of-the-art numerical black-box optimization techniques fail to capture the global structure of the problem, an important shortcoming that may guide their future development. We also provide a parallel implementation of the best-known algorithm to compute the discrepancy.

翻译：$L_{\infty}$星形偏差是衡量取自$[0,1)^d$的有限点集均匀性的一种度量。低偏差点集在数值积分中的拟蒙特卡洛方法及其他多种应用中具有高度相关性。遗憾的是，计算给定点集的$L_{\infty}$星形偏差被认为是一个难题，即使对于维度约为8的中等维数，现有最优精确算法也难以胜任。然而，尽管寻找定义该点集$L_{\infty}$星形偏差的全局最大值存在困难，在选定点处的局部评估代价较低。这使得该问题可通过黑箱优化方法处理。本研究在$L_{\infty}$星形偏差计算问题上比较了8种流行的数值黑箱优化算法，使用了维度从2到15的广泛实例集。我们发现，所有使用的优化器在绝大多数实例上表现极差，且在许多情况下随机搜索甚至优于更复杂的求解器。我们推测，当前最先进的数值黑箱优化技术未能捕捉到该问题的全局结构，这一重要不足可能指导其未来的发展。我们还提供了计算该偏差的已知最优算法的并行实现。