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Subdifferentials of convex functions and sigma-cyclic monotonicity

Published online by Cambridge University Press:  17 April 2009

Aris Daniilidis
Aris Daniilidis
Affiliation:
Laboratorie de Mathématiques Appliquées, CNRS UPRES A 5033, Université de Pau et des Pays de l'Adour, avenue de l'Université, 64000 Pau, France e-mail: [email protected]
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Abstract

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The property of σcyclic monotonicity is proposed here to describe subdifferentials of lsc convex functions that are continuous in their domains. It is shown that all monotone operators in R and all densely defined cyclically monotome operators in Rn share this property. Examples of a densely defined maximal cyclically monotone operator in a Hilbert space and of a subdifferential of a convex lsc function in R2 which are not σ-cyclically monotone operators are given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 2 , April 2000 , pp. 269 - 276
Copyright
Copyright © Australian Mathematical Society 2000

References

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]Clarke, F.H., Optimization and nonsmooth analysis (Wiley Interscience, New York, NY, 1983).Google Scholar
[3]Phelps, R., Convex functions, monotone operators and differentiability (2nd edition), Lecture Notes in Mathematics 1364 (Springer-Verlag, Berlin, 1991).Google Scholar
[4]Rockafellar, R.T., Convex analysis (Princeton University Press, Princeton NJ, 1970).CrossRefGoogle Scholar
[5]Rockafellar, R.T., ‘On the maximal monotonicity of subdifferential mappings’, Pacific J. Math 33 (1970), 209–216.CrossRefGoogle Scholar
[6]Rockafellar, R.T., ‘Generalized directional derivatives and subgradients of nonconvex functions’, Canad. J. Math 32 (1980), 257–280.CrossRefGoogle Scholar