Abstract

The subject of this paper concerns the remainder term in the first-order development of a (finite-valued) convex function. We study functions for which this term is comparable to a squared norm and we relate it to the corresponding remainder term of the conjugate function. We show that a convex function satisfies a quadratic growth condition if and only if its subdifferential satisfies a linear growth condition. Finally, we define a new concept of “tangential regularization”, involving a local decomposition of RN\mathbb{R}^N, along the subspace where the function is “smooth” and the subspace parallel to the subdifferential.

Author information

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

C. Lemaréchal

INRIA, BP 105, 78153 Le Chesnay, France.

C. Sagastizábal

INRIA, BP 105, 78153 Le Chesnay, France.

Keywords

  • Conjugate functions
  • first-order developments
  • regularization
  • growth conditions

References

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Suggested citation

C. Lemaréchal, C. Sagastizábal. “More Than First-Order Developments of Convex Functions: Primal-Dual Relations.” Journal of Convex Analysis 3 (1996), No. 2, 255–268. https://doi.org/10.68381/jca03017

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Published by Heldermann Verlag, 1996. Rights now held by Banach Press.