Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T05:03:51.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Dual characterizations of relative continuity of convex functions

Part of: Nonlinear operators and their properties Real functions General convexity

Published online by Cambridge University Press:  09 April 2009

J. Benoist  and
A. Daniilidis
J. Benoist
Affiliation:
LACO, CNRS UPRES 6090 Faculté des Sciences Université de Limoges123, avenue Albert Thomas 87060 Limoges, CedexFrance e-mail: [email protected]
A. Daniilidis
Affiliation:
CNRS ERS 2055 Laboratoire de Mathématiques Appliquées Université de Pau et des Pays de l'Adour avenue de l's Université 64000 PauFrance e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Various properties of continuity for the class of lower semicontinuous convex functions are considered and dual characterizations are established. In particular, it is shown that the restriction of a lower semicontinuous convex function to its domain (respectively, domain of subdifferentiability) is continuous if and only if its subdifferential is strongly cyclically monotone (respectively, σ-cyclically monotone).

Keywords

convexitycontinuitysubdifferentialcyclic monotonicity.

MSC classification

Secondary: 47H05: Monotone operators and generalizations 52A41: Convex functions and convex programs 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 2 , April 2001 , pp. 211 - 224
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Aussel, D., Corvellec, J.-N. and Lassonde, M., ‘Mean value property and subdifferential criteria for lower semicontinuous functions’, Trans. Amer. Math. Soc. 347 (1995), 4147–4161.CrossRefGoogle Scholar
[2]Borwein, J., Moors, W. and Shao, Y., ‘Subgradient representation of multifunctions’, J. Austral. Math. Soc. (Series B) 40 (1998), 1–13.Google Scholar
[3]Correa, R., Jofre, A. and Thibault, L., ‘Characterization of lower semicontinuous convex functions’, Proc. Amer. Math. Soc. 116 (1992), 67–72.CrossRefGoogle Scholar
[4]Daniilidis, A., ‘Subdifferentials of convex functions and sigma-cyclic monotonicity’, Bull. Austral. Math. Soc. 61 (2000), 269–276.CrossRefGoogle Scholar
[5]Daniilidis, A. and Hadjisavvas, N., ‘On the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity’, J. Math. Anal. Appl. 237 (1999), 30–42.CrossRefGoogle Scholar
[6]Phelps, R., Convex functions, monotone operators and differentiability, 2nd edition (Springer, Berlin, 1991).Google Scholar
[7]Rockafellar, R. T., Convex analysis (Princeton University Press, Princeton NJ, 1970).CrossRefGoogle Scholar
[8]Rockafellar, R. T., ‘On the maximal monotonicity of subdifferential mappings’, Pacific J. Math. 33 (1970), 209–216.CrossRefGoogle Scholar