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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: joel.benoist@unilim.fr
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: aris.daniilidis@univ-pau.fr
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Abstract

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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