ABSTRACT

We generalize the existing formulation and results on linear separability of sets. In order to characterize the solution of the generalized problem, we use the concepts of convex hulls. For finite sets, it is well known the Support Vector Machine technique for finding the optimal separating hyperplane. Here we consider arbitrary sets, allowing infinite, unbounded and nonclosed sets. The problem is formulated as an optimization problem with possibly infinitely many constraints. We prove existence and uniqueness of the solution. Besides, we present some examples and counterexamples to many properties discussed in the text and statements in the literature.

中文翻译：

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关于任意集合的最优分离超平面：SVM 公式的推广和凸包方法

摘要

我们将现有的公式和结果推广到集合的线性可分性上。为了表征广义问题的解决方案，我们使用凸包的概念。对于有限集，众所周知的支持向量机技术用于寻找最优分离超平面。在这里，我们考虑任意集合，允许无限、无界和非闭集。该问题被表述为具有可能无限多约束的优化问题。我们证明了解的存在性和唯一性。此外，我们对文本中讨论的许多属性和文献中的陈述提出了一些例子和反例。