Journal of Convex Analysis 14 (2007), No. 1, 185--204
Copyright Heldermann Verlag 2007
Convex Along Lines Functions and Abstract Convexity. Part I
Giovanni P. Crespi
University of the Aosta Valley, Faculty of Economics, 11100 Aosta, Italy
g.crespi@univda.it
Ivan Ginchev
Technical University, Dept. of Mathematics, 9010 Varna, Bulgaria
ginchev@ms3.tu-varna.acad.bg
Matteo Rocca
University of Insubria, Dept. of Economics, 21100 Varese, Italy
mrocca@eco.uninsubria.it
Alexander Rubinov
University of Ballarat, Center for Informatics and Applied Optimization, P. O. Box 663, Ballarat, Australia
a.rubinov@ballarat.edu.au
[Abstract-pdf]
\def\RR{\mathbb R} \def\rc#1{{\rm #1\,}} \newcommand{\calH}{{\cal H}} \newcommand{\calL}{{\cal L}} The present paper investigates the property of a function $f\colon \RR^n \to \RR_{+\infty} := \RR \cup \{+\infty\}$ with $f(0) < +\infty$ to be ${\cal L}_n$-subdifferentiable or $\calH_n$-convex. The $\calL_n$-subdifferentiability and $\calH_n$-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 $\calL_n^0$-subdifferentiability and $\calH_n^0$-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 $f\colon \RR^n \to \RR_{+\infty}$ be such that $f(0) < +\infty$ and $f$ is $\calH_n$-convex at the points at which it is infinite. Then if $f$ is $\calL_n^0$-subdifferentiable, it is CAL and globally calm at each $x^0\in\rc{dom}f$. 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 $\calH_n^0$-convex (Theorems 6.3and 6.4).
Keywords: Abstract convexity, generalized convexity, duality, H-n-convexity, convex-along-rays functions, convex-along-lines functions, positively homogeneous functions.
MSC: 49J52, 49N15
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