Clipping algorithms for solving the nearest point problem over reduced convex hulls

Abstract

The nearest point problem (NPP), i.e., finding the closest points between two disjoint convex hulls, has two classical solutions, the Gilbert–Schlesinger–Kozinec (GSK) and Mitchell–Dem’yanov–Malozemov (MDM) algorithms. When the convex hulls do intersect, NPP has to be stated in terms of reduced convex hulls (RCHs), made up of convex pattern combinations whose coefficients are bound by a μ < 1 value and that are disjoint for suitable μ . The GSK and MDM methods have recently been extended to solve NPP for RCHs using the particular structure of the extreme points of a RCH. While effective, their reliance on extreme points may make them computationally costly, particularly when applied in a kernel setting. In this work we propose an alternative clipped extension of classical MDM that results in a simpler algorithm with the same classification accuracy than that of the extensions already mentioned, but also with a much faster numerical convergence.