Robust Set Separation Via Exponentials

Given a pair of finite disjoint sets A and B in Euclidean n-space, a fundamental problem with numerous applications is to efficiently determine a hyperplane H(ω, γ) which separates these sets when they are separable, or 'nearly' separates them when they are not. We seek a hyperplane that separates them in the sense that a measure of the Euclidean distance between the separating hyperplane and all of the points is as large as possible. This is done by 'weighting' points relative to A ∪ B according to their distance to H(ω, γ), with the closer points getting a higher weight, but still taking into account the points distant from H(ω, γ). The negative exponential is chosen for that purpose. In this paper we examine the optimization problem associated with this set separation problem and characterize it (convex or non-convex).