Skip to main content
×
×
Home

Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory

  • J. M. Borwein (a1) and H. M. Strojwas (a2)
Extract

As various types of tangent cones, generalized derivatives and subgradients prove to be a useful tool in nonsmooth optimization and nonsmooth analysis, we witness a considerable interest in analysis of their properties, relations and applications.

Recently, Treiman [18] proved that the Clarke tangent cone at a point to a closed subset of a Banach space contains the limit inferior of the contingent cones to the set at neighbouring points. We provide a considerable strengthening of this result for reflexive spaces. Exploring the analogous inclusion in which the contingent cones are replaced by pseudocontingent cones we have observed that it does not hold any longer in a general Banach space, however it does in reflexive spaces. Among the several basic relations we have discovered is the following one: the Clarke tangent cone at a point to a closed subset of a reflexive Banach space is equal to the limit inferior of the weak (pseudo) contingent cones to the set at neighbouring points.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory
      Available formats
      ×
Copyright
References
Hide All
1. Bishop, E. and Phelps, R. R., The support Junctionals of a convex set, Proc. Symp. Amer. Math. Soc. 139 (1963), 433467.
2. Borwein, J. M., Tangent cones, starshape and convexity, Internat. J. Math. & Math. Sci. 1 (1978), 459477.
3. Borwein, J. M., Weak local sup port ability and applications to approximation, Pacific Journal of Mathematics 82 (1979), 323338.
4. Borwein, J. M. and O'Brien, R., Tangent cones and convexity, Canadian Math. Bull. 19 (1976), 257261.
5. Borwein, J. M. and Strojwas, H., Tangential approximations, to appear in Nonlinear Anal, Theory, Meth. Appl.
6. Clarke, F. H., Optimization and nonsmooth analysis (John Wiley, 1982).
7. Cornet, B., Regular properties of tangent and normal cones, (to appear).
8. Diestel, J., Geometry of Banach spaces — selected topics (Springer-Verlag, 1975).
9. Ekeland, I., On the variational principle, J. Math. Anal. Appl. 47 (1974), 324353.
10. Holmes, R. B., Geometric functional analysis and its applications (Springer-Verlag, 1975).
11. Ioffe, A. D., Approximate subdifferentials and applications. 1: The finite dimensional theory, Trans, of the Amer. Math. Soc. 281 (1984), 389416.
12. Krasnoselski, M., Sur un critère pour qu'un domaine soit étoileé, Rec. Math. [Math Sbornik] N.S. 19 (1946), 309310.
13. Lau, K., Almost Chebyshev subsets in reflexive Banach spaces, Indianna Univ. Math. J. 27 (1978), 791795.
14. Penot, J. P., A characterization of tangential regularity, Nonlinear Anal., Theory, Meth. Appl. 5 (1981), 625663.
15. Penot, J. P., A characterization of Clarke's strict tangent cone via nonlinear semigroup, (to appear).
16. Penot, J. P., The Clarke's tangent cone and limits of tangent cones, (to appear).
17. Rockafellar, R. T., Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math, of Operations Research 6 (1981), 424436.
18. Treiman, J. S., A new characterization of Clarke's tangent cone and its applications to subgradient analysis and optimization, Ph.D. Thesis, Univ. of Washington, (1983).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed