Geodesic
Conic Separation of Finite Sets. I: The homogeneous case
Journal of convex analysis, Tome 21 (2014) no. 1, pp. 1-28

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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.
Classification : 90C25, 90C26
Mots-clés : Conical separation, revolution cone, convex optimization, DC-optimization, proximal point techniques, classification 
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     title = {Conic {Separation} of {Finite} {Sets.} {I:} {The} homogeneous case},
     journal = {Journal of convex analysis},
     pages = {1--28},
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     url = {http://geodesic.mathdoc.fr/item/JCA_2014_21_1_JCA_2014_21_1_a0/}
}
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A. Astorino; M. Gaudioso; A. Seeger. Conic Separation of Finite Sets. I: The homogeneous case. Journal of convex analysis, Tome 21 (2014) no. 1, pp. 1-28. http://geodesic.mathdoc.fr/item/JCA_2014_21_1_JCA_2014_21_1_a0/