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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

Ivan Ginchev
Technical University, Dept. of Mathematics, 9010 Varna, Bulgaria

Matteo Rocca
University of Insubria, Dept. of Economics, 21100 Varese, Italy

Alexander Rubinov
University of Ballarat, Center for Informatics and Applied Optimization, P. O. Box 663, Ballarat, Australia


\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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