Determining an upper bound, particularly the optimal upper bound of the Gini coefficient when dealing with grouped data without specified income brackets, remains an important and open question. In this paper, we introduce an efficient algorithm to calculate the exact optimal upper bound of the Gini coefficient with provable guarantees. To exemplify these methods, we also offer computed results for the Gini coefficients of urban and rural China spanning the years 2003 to 2008.

翻译：在未指定收入分组边界的情况下，如何确定基尼系数的上界，特别是最优上界，仍然是一个重要且尚未完全解决的问题。本文提出了一种高效算法，用于计算具有可证明保证的基尼系数精确最优上界。为验证该方法，我们还提供了2003年至2008年中国城镇与农村地区基尼系数的计算结果。