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

• Autorzy: Stein, O.
• Języki publikacji: EN

Abstrakty

EN

We 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.

Słowa kluczowe

EN

convexity; strict convexity; strong convexity; true convexity; differentiable characterization; full dimensional convex set

• Czasopismo: Schedae Informaticae
• Rocznik: 2012
• Tom: Vol. 21
• Strony: 55--63
• Opis fizyczny: Bibliogr. 5 poz.

Twórcy

autor

Stein, O.

• Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Kaiserstr. 12, 76131 Karlsruhe, Germany,

Bibliografia

• [1] Hiriart-Urruty J.-B., Lemaréchal C.; Fundamentals of Convex Analysis, Springer 2001.
• [2] Jongen H.Th., Schmidt P.G.; Analysis, Lecture Notes, Aachener Beiträge zur Mathematik 19, Wissenschaftsverlag, Mainz 1998.
• [3] Rockafellar R.T.; Convex Analysis, Princeton University Press 1970.
• [4] Rockafellar R.T., Wets R.; Variational Analysis, Springer 1998.
• [5] Shikhman V., Stein O.; On jet convex functions and their tensor products, Optimization 61, 2012, pp. 717–731.