DOI: 10.68381/jca14014 ISSN: 0944-6532
Convex Along Lines Functions and Abstract Convexity. Part I
Giovanni P. Crespi, Ivan Ginchev, Matteo Rocca, Alexander Rubinov
The present paper investigates the property of a function
f\colon \mathbb{R}^n \to \mathbb{R}_{+\infty}:= \mathbb{R} \cup \{+\infty\}
f
:
R
n
→
R
+
∞
:
=
R
∪
{
+
∞
}
with
f(0) < +\infty
f
(
0
)
<
+
∞
to be
{\cal L}_n
L
n
-subdifferentiable or
\mathcal{H}_n
H
n
-convex. The
\mathcal{L}_n
L
n
-subdifferentiability and
\mathcal{H}_n
H
n
-convexity are introduced as in the book of A. M. Rubinov [“Abstract convexity and global optimization”, Kluwer Academic Publishers, Dordrecht 2000]. Some refinements of these properties lead to the notions of
\mathcal{L}_n^0
L
n
0
-subdifferentiability and
\mathcal{H}_n^0
H
n
0
-convexity. Their relation to the convex-along (CAL) functions is underlined in the following theorem proved in the paper (Theorem 5.2): Let the function
f\colon \mathbb{R}^n \to \mathbb{R}_{+\infty}
f
:
R
n
→
R
+
∞
be such that
f(0) < +\infty
f
(
0
)
<
+
∞
and
f
f
is
\mathcal{H}_n
H
n
-convex at the points at which it is infinite. Then if
f
f
is
\mathcal{L}_n^0
L
n
0
-subdifferentiable, it is CAL and globally calm at each
x^0\in\operatorname{dom}f
x
0
∈
dom
f
. Here the notions of local and global calmness are introduced after R. T. Rockafellar and R. J-B Wets [“Variational analysis”, Springer-Verlag, Berlin 1998] and play an important role in the considerations. The question is posed for the possible reversal of this result. In the case of a positively homogeneous (PH) and CAL function such a reversal is proved (Theorems 6.2). As an application conditions are obtained under which a CAL PH function is
\mathcal{H}_n^0
H
n
0
-convex (Theorems 6.3and 6.4).