Journal of Convex Analysis 21 (2014), No. 1, 001--028
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@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
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