Abstract

The present paper investigates the property of a function f ⁣:Rn→R+∞:=R∪{+∞}f\colon \mathbb{R}^n \to \mathbb{R}_{+\infty}:= \mathbb{R} \cup \{+\infty\} with f(0)<+∞f(0) < +\infty to be Ln{\cal L}_n-subdifferentiable or Hn\mathcal{H}_n-convex. The Ln\mathcal{L}_n-subdifferentiability and Hn\mathcal{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 Ln0\mathcal{L}_n^0-subdifferentiability and Hn0\mathcal{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 ⁣:Rn→R+∞f\colon \mathbb{R}^n \to \mathbb{R}_{+\infty} be such that f(0)<+∞f(0) < +\infty and ff is Hn\mathcal{H}_n-convex at the points at which it is infinite. Then if ff is Ln0\mathcal{L}_n^0-subdifferentiable, it is CAL and globally calm at each x0∈dom⁡fx^0\in\operatorname{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 Hn0\mathcal{H}_n^0-convex (Theorems 6.3and 6.4).

Author information

Contact details are reproduced from the original publication and may be historical.

Giovanni P. Crespi

University of the Aosta Valley, Faculty of Economics, 11100 Aosta, Italy

g.crespi@univda.it

Alexander Rubinov

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

a.rubinov@ballarat.edu.au

Keywords

  • Abstract convexity
  • generalized convexity
  • duality
  • H-n-convexity
  • convex-along-rays functions
  • convex-along-lines functions
  • positively homogeneous functions

Mathematics Subject Classification

49J52, 49N15

References

  1. J. M. Borwein, A. S. Lewis: Convex Analysis and Nonlinear Optimization. Theory and Examples, Springer, Berlin (2000).https://doi.org/10.1007/978-1-4757-9859-3Google Scholar
  2. J. Dutta, J.-E. Martinez-Legaz, A. Rubinov: Monotonic analysis over cones, Optimization, to appear.Google Scholar
  3. I. Ginchev, V. I. Ivanov: Second-order characterizations of convex and pseudoconvex functions, J. Appl. Anal. 9(2) (2003) 261–273.https://doi.org/10.1515/JAA.2003.261Google Scholar
  4. J.-J. Moreau: Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl. 49 (1970) 109–154.Google Scholar
  5. D. Pallaschke, S. Rolewicz: Foundation of Mathematical Optimization (Convex Analysis without Linearity), Kluwer Academic Publishers, Dordrecht (1997).https://doi.org/10.1007/978-94-017-1588-1Google Scholar
  6. J.-P. Penot: A duality for starshaped functions, Bull. Pol. Acad. Sci., Math. 50(2) (2002) 127–139.Google Scholar
  7. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  8. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  9. A. M. Rubinov: Abstract Convexity and Global Optimization, Kluwer Academic Publishers, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9Google Scholar
  10. A. Rubinov, Z. Dzalilov: Abstract convexity of positively homogeneous functions, J. Stat. Manag. Syst. 5(1–3) (2002) 1–20.https://doi.org/10.1080/09720510.2002.10701048Google Scholar
  11. A. P. Shveidel: Separability of star-shaped sets and its application to an optimization problem, Optimization 40(3) (1997) 207–227.https://doi.org/10.1080/02331939708844310Google Scholar
  12. A. P. Shveidel: Abstract convex sets with respect to the class of general min-type functions, Optimization 52 (2003) 571–580.https://doi.org/10.1080/02331930310001611448Google Scholar
  13. I. Singer: Abstract Convex Analysis, Wiley-Interscience, New York (1997).Google Scholar

Suggested citation

G. P. Crespi, I. Ginchev, M. Rocca, A. Rubinov. “Convex Along Lines Functions and Abstract Convexity. Part I.” Journal of Convex Analysis 14 (2007), No. 1, 185–204. https://doi.org/10.68381/jca14014

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.