Journal of Convex Analysis 21 (2014), No. 3, 819--831
Copyright Heldermann Verlag 2014
Conic Separation of Finite Sets. II: The Non-Homogeneous Case
Annabella Astorino
Istituto di Calcolo e Reti ad Alte Prestazioni C.N.R., Dip. di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, 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@dimes.unical.it
Alberto Seeger
Dept. of Mathematics, University of Avignon, 33 rue Louis Pasteur, 84000 Avignon, France
alberto.seeger@univ-avignon.fr
[Abstract-pdf]
[For part I of this article see this journal 21 (2013), Number 1.]\par We address the issue of separating two finite sets in $\mathbb{R}^n $ by means of a suitable revolution cone $$ \Gamma (z,y,s)= \{x \in \mathbb{R}^n :\, s\,\Vert x-z\Vert - y^T(x-z)=0\}. $$ One has to select the aperture coefficient $s$, the axis $y$, and the apex $z$ in such a way as to meet certain optimal separation criteria. The homogeneous case $z=0$ has been treated in Part I of this work. We now discuss the more general case in which the apex of the cone is allowed to move in a certain region. The non-homogeneous case is structurally more involved and leads to challenging nonconvex nonsmooth optimization problems.
Keywords: Conical separation, revolution cone, alternating minimization, DC programming, classification.
MSC: 90C26
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