Abstract

For x ∈ H ∖ S and δ ≥ 0, the δ-projection of x onto S, is the set projSδ(x):={s∈S:∥s−x∥2≤dS(x)2+δ2}\mathrm{proj}^{\delta}_{S}(x) := \{s \in S : \|s - x\|^2 \leq d_S(x)^2 + \delta^2\}. We prove that each vector x − s with s∈projSδ(x)s \in \mathrm{proj}^{\delta}_{S}(x) can be approximated by some nearby proximal normal. We also give a simple proof (new in the context of an infinite dimensional Hilbert space) of a result due to Rockafellar [17] concerning the approximation of “horizontal” normals to the epigraph of a lower semicontinuous function by “non-horizontal” ones.

Author information

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

M. L. Radulescu

Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada.

F. H. Clarke

Centre de recherches mathématiques, Université de Montréal, C. P.6128, succ. Centre-ville, Montréal QC H3C 3J7, Canada

Keywords

  • Nonsmooth analysis
  • distance function
  • δ-projection
  • proximal normal
  • proximal subdifferential

References

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Suggested citation

M. L. Radulescu, F. H. Clarke. “Geometric Approximation of Proximal Normals.” Journal of Convex Analysis 4 (1997), No. 2, 373–379. https://doi.org/10.68381/jca04022

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Published by Heldermann Verlag, 1997. Rights now held by Banach Press LLC.