On the optimal separating hyperplane for arbitrary sets: a generalization of the SVM formulation and a convex hull approach. (2nd January 2022)

• Record Type: Journal Article
• Authors: Ribeiro, Ademir A.; Sachine, Mael
• Is Part Of: Optimization. Volume 71:Number 1(2022)
• Page Start: 213
• Page End: 226
• Publication Date: 2022-01-02
• Subjects: Linear separability -- convex hull -- optimal separating hyperplane -- arbitrary sets -- support vector machines -- infinite sets; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2020.1830089
• Languages: English
• ISSNs: 0233-1934

Abstract

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.