
Journal of Convex Analysis 21 (2014), No. 1, 001028 Copyright Heldermann Verlag 2014 Conic Separation of Finite Sets. I: The homogeneous case Annabella Astorino Istituto di Calcolo e Reti ad Alte Prestazioni C.N.R., Università della Calabria, 87036 Rende, Italy astorino@icar.cnr.it Manlio Gaudioso Dip. di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, Università della Calabria, 87036 Rende, Italy gaudioso@deis.unical.it Alberto Seeger Dept. of Mathematics, University of Avignon, 33 rue Louis Pasteur, 84000 Avignon, France alberto.seeger@univavignon.fr [Abstractpdf] This work addresses the issue of separating two finite sets in $\mathbb{R}^n$ by means of a suitable revolution cone $$ \Gamma (z,y,s)= \{x \in \mathbb{R}^n : s\,\Vert xz\Vert  y^T(xz)=0\}. $$ The specific challenge at hand is to determine the aperture coefficient $s$, the axis $y$, and the apex $z$ of the cone. These parameters have to be selected in such a way as to meet certain optimal separation criteria. Part I of this work focusses on the homogeneous case in which the apex of the revolution cone is the origin of the space. The homogeneous case deserves a separated treatment, not just because of its intrinsic interest, but also because it helps to built up the general theory. Part II of this work concerns the nonhomogeneous case in which the apex of the cone can move in some admissible region. The nonhomogeneous case is structurally more involved and leads to challenging nonconvex nonsmooth optimization problems. Keywords: Conical separation, revolution cone, convex optimization, DCoptimization, proximal point techniques, classification. MSC: 90C25, 90C26 [ Fulltextpdf (214 KB)] for subscribers only. 