Abstract
The notions of convexificator and minimal convexificator of a posotively homogeneous (p.h.) function h : ℝn → ℝ are used to get a characterization of the convexity and concavity of h. It is shown that the uniqueness of a minimal (by inclusion) convexificator is a necessary and sufficient condition for a p.h. function to be convex or concave. Representations of a p.h. function in terms of its minimal convexificators are also derived.
This research was supported by the Russian Foundation for Fundamental Studies under Grant RFFI No. 97-01-00499
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Demyanov, V.F., Rubinov, A.M. (1995), Constructive Nonsmooth Analysis, Verlag Peter Lang, Franfurt a/M.
Rockafellar, R.T. (1970), Convex Analysis, Princeton University Press, Princeton N.J.
Pschenichnyi, B.N. (1980), Convex Analysis and Extremal Problems, Nauka, Moscow.
Clarke, F.M. (1983), Optimization and Nonsmooth Analysis,Wiley Intersciencem New York.
Michel, P., Penot, J.-P. (1984), Calcus sous-differential pour les fonctions lipschitzienness et non-lipschitziennes, C.R. Acad. Sc. Paris, Ser. 1298, pp. 269 - 272.
Demyanov, V.F. (1994), Convexification and concavification of a positively homogeneous function by the same family of linear functions, Universita di Pisa, Report 3,208, 802.
Demyanov, V.F., Jeyakumar, V. (1997), Hunting for a smaller convex subdifferential, J. of Global Optimization, Vol, 10, No. 3, pp. 305 - 326.
Jeyakumar, V., Demyanov, V.F. (1996), A mean-value theorem and a characterization of convexity using convexificators, Applied Math. Research Report AMR 96/13, Univ. of New South Wales, Sydney, Australia.
Jeyakumar, V., Luc, D.T., Schaible, S. (1998), Characterizations of generalized monotone nonsmooth continuous maps using Approximate Jacobians, J. of Convex Analysis, Vol. 5, No. 1, pp. 119 - 132.
10. Demyanov, V.F., Murzabekova, G.E. (1999), Convexificators and Implicit functions in Nonsmooth Systems, Forthcoming in J. of Comput. Math. and Math. Physics.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Demyanov, V.F. (2000). Minimal Convexificators of a positively Homogeneous Function and a Characterization of Its Convexity and Concavity. In: Pillo, G.D., Giannessi, F. (eds) Nonlinear Optimization and Related Topics. Applied Optimization, vol 36. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3226-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3226-9_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4823-6
Online ISBN: 978-1-4757-3226-9
eBook Packages: Springer Book Archive