- Title
- Convex along lines functions and abstract convexity. Part i
- Creator
- Crespi, G. P.; Ginchev, I.; Rocca, M.; Rubinov, Alex
- Date
- 2007
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/34012
- Identifier
- vital:216
- Identifier
- ISSN:0944-6532
- Abstract
- The present paper investigates the property of a function f : Rn → R+∞ := R U {+∞} with f(0) < +∞ to be Ln-subdifferentiable or Hn-convex. The Ln-subdifferentiability and Hnn-convexity are introduced as in Rubinov [9]. Some refinements of these properties lead to the notions of Ln0-subdifferentiability and Hn0-convexity. Their relation to the convex-along (CAL) functions is underlined in the following theorem proved in the paper (Theorem 5.6): Let the function f : Rn → R+∞ be such that f(0) < +∞ and f is Hn-convex at the points at which it is infinite. Then if f is Ln0-subdifferentiable, it is CAL and globally calm at each x0 ∈ dom f. Here the notions of local and global calmness are introduced after Rockafellar, Wets [8] 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 (Theorem 6.2). As an application conditions are obtained under which a CAL PH function is Hn0-convex (Theorems 6.3 and 6.4). © Heldermann Verlag.; C1
- Publisher
- Heldermann Verlag
- Relation
- Journal of Convex Analysis Vol. 14, no. 1 (2007), p. 185-204
- Rights
- Centre for Informatics and Applied Optimization
- Rights
- Copyright Heldermann Verlag
- Rights
- This metadata is freely available under a CCO license
- Subject
- 0101 Pure Mathematics; Abstract convexity; Convex-along-lines functions; Convex-along-rays functions; Duality; Generalized convexity; Positively homogeneous functions
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