Abstract

Every lower semicontinuous convex function can be represented through its subdifferential by means of an "integration" formula introduced by R. T. Rockafellar [Pacific J. Math. 33 (1970) 209–216]. We show that in Banach spaces with the Radon-Nikodym property this formula can be significantly refined under a standard coercivity assumption. This yields an interesting application to the convexification of lower semicontinuous functions.

Author information

Contact details are reproduced from the original publication and may be historical.

Joël Benoist

Faculté des Sciences, LACO, URA CNRS 1586, 123 av. Albert Thomas, Université de Limoges, France

benoist@unilim.fr

Aris Daniilidis

Dep. de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

arisd@mat.uab.es

Keywords

  • Convex function
  • subdifferential
  • epi-pointed function
  • cusco mapping
  • strongly exposed point

Mathematics Subject Classification

52A41, 46B22, 26E25

References

  1. M. Bachir, A. Daniilidis, J.-P. Penot: Lower subdifferentiability and integration, Set-Valued Anal. 10 (2002) 89–108.https://doi.org/10.1023/A:1014460029093Google Scholar
  2. J. Benoist, A. Daniilidis: Integration of Fenchel subdifferentials of epi-pointed functions, SIAM J. Optim. 12 (2002) 575–582.https://doi.org/10.1137/S0152623400381279Google Scholar
  3. J. Benoist, J.-B. Hiriart-Urruty: What is the subdifferential of the closed convex hull of a function?, SIAM J. Math. Anal. 27 (1996) 1661–1679.https://doi.org/10.1137/S0036141094265936Google Scholar
  4. J. M. Borwein: Minimal cuscos and subgradients of Lipschitz functions, in: Fixed Point Theory and its Applications, J.-B. Baillon, M. Théra (eds.), Pitman Res. Notes in Math. Series 252, Longman, Essex (1991) 57–82.Google Scholar
  5. J. Borwein, Q. Zhu: Multifunctional and functional analytic techniques in nonsmooth analysis, in: Nonlinear Analysis, Differential Equations and Control, F. H. Clarke, R. J. Stern (eds.), Series C: Mathematical and Physical Sciences 528, Kluwer Acad. Publ. (1999) 61–157.https://doi.org/10.1007/978-94-011-4560-2_2Google Scholar
  6. J. Borwein, W. Moors: Essentially smooth Lipschitz functions, J. Funct. Anal. 49 (1997) 305–351.https://doi.org/10.1006/jfan.1997.3101Google Scholar
  7. J. Collier: The dual of a space with the Radon-Nikodym property, Pacific J. Math. 64 (1976) 103–106.https://doi.org/10.2140/pjm.1976.64.103Google Scholar
  8. J.-B. Hiriart-Urruty, C. Lemarechal: Fundamentals of Convex Analysis, Grundlehren Text Editions, Springer (2001).https://doi.org/10.1007/978-3-642-56468-0Google Scholar
  9. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd Edition, Springer, Berlin (1991).Google Scholar
  10. R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970) 209–216.https://doi.org/10.2140/pjm.1970.33.209Google Scholar

Suggested citation

J. Benoist, A. Daniilidis. “Subdifferential Representation of Convex Functions: Refinements and Applications.” Journal of Convex Analysis 12 (2005), No. 2, 255–265. https://doi.org/10.68381/jca12018

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Published by Heldermann Verlag, 2005. Rights now held by Banach Press LLC.