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Twice Differentiable Characterizations of Convexity Notions for Functions on Full Dimensional Convex Sets

Publication date: 20.12.2012

Schedae Informaticae, 2012, Volume 21, pp. 55 - 63

https://doi.org/10.4467/20838476SI.12.004.0814

Authors

Oliver Stein
Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
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Titles

Twice Differentiable Characterizations of Convexity Notions for Functions on Full Dimensional Convex Sets

Abstract

IWe derive C2 −characterizations for convex, strictly convex, as well as strongly convex functions on full dimensional convex sets. In the cases of convex and strongly convex functions this weakens the well-known openness assumption on the convex sets. We also show that, in a certain sense, the full dimensionality assumption cannot be weakened further. In the case of strictly convex functions we weaken the well-known sufficient C2 −condition for strict convexity to a characterization. Several examples illustrate the results.

References

Hiriart-Urruty J.-B., Lemar´echal C.; Fundamentals of Convex Analysis, Springer 2001.

Jongen H.Th., Schmidt P.G.; Analysis, Lecture Notes, Aachener Beitr¨age zur Mathematik 19, Wissenschaftsverlag, Mainz 1998.

Rockafellar R.T.; Convex Analysis, Princeton University Press 1970.

Rockafellar R.T., Wets R.; Variational Analysis, Springer 1998.

Shikhman V., Stein O.; On jet convex functions and their tensor products, Optimization 61, 2012, pp. 717–731.

Information

Information: Schedae Informaticae, 2012, Volume 21, pp. 55 - 63

Article type: Original article

Titles:

Polish:

Twice Differentiable Characterizations of Convexity Notions for Functions on Full Dimensional Convex Sets

English:

Twice Differentiable Characterizations of Convexity Notions for Functions on Full Dimensional Convex Sets

Authors

Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

Published at: 20.12.2012

Article status: Open

Licence: None

Percentage share of authors:

Oliver Stein (Author) - 100%

Article corrections:

-

Publication languages:

English