DOI: 10.68381/jca21044 ISSN: 0944-6532
Conic Separation of Finite Sets. II: The Non-Homogeneous Case
Annabella Astorino, Manlio Gaudioso, Alberto Seeger
[For part I of this article see this journal 21 (2013), Number 1.] We address 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
}
.
One has to select the aperture coefficient
s
s
, the axis
y
y
, and the apex
z
z
in such a way as to meet certain optimal separation criteria. The homogeneous case
z=0
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.