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Conic separation of finite sets:The homogeneous case

Annabella Astorino(astorino***at***icar.cnr.it)
Manlio Gaudioso(gaudioso***at***deis.unical.it)
Alberto Seeger(alberto.seeger***at***univ-avignon.fr)

Abstract: 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.

Category 1: Applications -- Science and Engineering

Category 2: Convex and Nonsmooth Optimization

Citation: 90C25, 90C26

Download: [PDF]

Entry Submitted: 10/14/2013
Entry Accepted: 10/14/2013
Entry Last Modified: 10/14/2013

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