Abstract

For finite-valued convex functions ff defined on the nn-dimensional Euclidean space, we are interested in the set-valued mapping assigning to each pair (f,x)(f,x) the subdifferential of ff at xx. Our approach is uniform with respect to ff in the sense that it involves pairs of functions close enough to each other, but not necessarily around a nominal function. More precisely, we provide lower and upper estimates, in terms of Hausdorff excesses, of the subdifferential of one of such functions at a nominal point in terms of the subdifferential of nearby functions in a ball centered in such a point. In particular, we obtain the (1/2) - Hölder calmness of our mapping at a nominal pair (f,x)(f,x) under the assumption that the subdifferential mapping viewed as a set-valued mapping from Rn\mathbb{R}^{n} to Rn\mathbb{R}^{n} with ff fixed is calm at each point of {x}×∂f(x)\{x\}\times \partial f(x).

Author information

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

María Josefa Cánovas

Center of Operations Research, Miguel Hernández University, 03202 Elche, Spain

canovas@umh.es

Marco Antonio López

Department of Mathematics, University of Alicante, 03071 Alicante, Spain
and: CIAO, Federation University, Australia

marco.antonio@ua.es

Juan Parra

Center of Operations Research, Miguel Hernández University, 03202 Elche, Spain

parra@umh.es

Keywords

  • Convex functions
  • sudifferentials
  • Hausdorff excess
  • uniform spaces
  • Hölder calmness

Mathematics Subject Classification

49J53, 52A41, 90C25, 90C31, 90C34

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Suggested citation

G. Beer, M. J. Cánovas, M. A. López, J. Parra. “A Uniform Approach to Hölder Calmness of Subdifferentials.” Journal of Convex Analysis 27 (2020), No. 1, 165–178. https://doi.org/10.68381/jca27010

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press.