DOI: 10.68381/jca21001 ISSN: 0944-6532
Conic Separation of Finite Sets. I: The homogeneous case
Annabella Astorino, Manlio Gaudioso, Alberto Seeger
This work addresses the issue of separating two finite sets in
\mathbb{R}^n
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\}.
Γ
(
z
,
y
,
s
)
=
{
x
∈
R
n
:
s
∥
x
−
z
∥
−
y
T
(
x
−
z
)
=
0
}
.
The specific challenge at hand is to determine the aperture coefficient
s
s
, the axis
y
y
, and the apex
z
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.