We introduce and develop a notion of “catastrophic radii” to identify where a minimization method may require an arbitrarily large number of steps to approximate a minimizer of an objective function, and we use this notion to categorize the performance of method/objective combinations. In order to investigate the different categories, we explore simple examples where explicit formulas can be used, and we discuss several ways that simulation can be used to investigate catastrophic radii for other method/objective combinations.

中文翻译：

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在数值最小化中使用灾难性半径进行分类

我们引入并发展了“灾难性半径”的概念，以识别最小化方法可能在何处需要大量步骤来逼近目标函数的最小化器，并使用该概念对方法/目标组合的性能进行分类。为了研究不同的类别，我们探索了可以使用显式公式的简单示例，并讨论了可用于模拟研究其他方法/目标组合的灾难性半径的几种方法。