Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Journal of Convex Analysis 21 (2014), No. 1, 001--028Copyright 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@univ-avignon.fr [Abstract-pdf] This work addresses the issue of separating two finite sets in $\mathbb{R}^n$ by means of a suitable revolution cone $$\Gamma (z,y,s)= \{x \in \mathbb{R}^n : s\,\Vert x-z\Vert - y^T(x-z)=0\}.$$ 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 non-homogeneous case in which the apex of the cone can move in some admissible region. The non-homogeneous case is structurally more involved and leads to challenging nonconvex nonsmooth optimization problems. Keywords: Conical separation, revolution cone, convex optimization, DC-optimization, proximal point techniques, classification. MSC: 90C25, 90C26 [ Fulltext-pdf  (214  KB)] for subscribers only.