Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T06:54:24.350Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 November 2018

J. M. Borwein  and
H. M. Strojwas
J. M. Borwein
Affiliation:
Dalhousie University, Halifax, Nova Scotia
H. M. Strojwas
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 2 , 01 April 1986 , pp. 431 - 452
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Bishop, E. and Phelps, R. R., The support Junctionals of a convex set, Proc. Symp. Amer. Math. Soc. 139 (1963), 433–467.Google Scholar
2. Borwein, J. M., Tangent cones, starshape and convexity, Internat. J. Math. & Math. Sci. 1 (1978), 459–477.Google Scholar
3. Borwein, J. M., Weak local sup port ability and applications to approximation, Pacific Journal of Mathematics 82 (1979), 323–338.Google Scholar
4. Borwein, J. M. and O'Brien, R., Tangent cones and convexity, Canadian Math. Bull. 19 (1976), 257–261.Google Scholar
5. Borwein, J. M. and Strojwas, H., Tangential approximations, to appear in Nonlinear Anal, Theory, Meth. Appl.Google Scholar
6. Clarke, F. H., Optimization and nonsmooth analysis (John Wiley, 1982).Google Scholar
7. Cornet, B., Regular properties of tangent and normal cones, (to appear).Google Scholar
8. Diestel, J., Geometry of Banach spaces — selected topics (Springer-Verlag, 1975).CrossRefGoogle Scholar
9. Ekeland, I., On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.Google Scholar
10. Holmes, R. B., Geometric functional analysis and its applications (Springer-Verlag, 1975).CrossRefGoogle Scholar
11. Ioffe, A. D., Approximate subdifferentials and applications. 1: The finite dimensional theory, Trans, of the Amer. Math. Soc. 281 (1984), 389–416.Google Scholar
12. Krasnoselski, M., Sur un critère pour qu'un domaine soit étoileé, Rec. Math. [Math Sbornik] N.S. 19 (1946), 309–310.Google Scholar
13. Lau, K., Almost Chebyshev subsets in reflexive Banach spaces, Indianna Univ. Math. J. 27 (1978), 791–795.Google Scholar
14. Penot, J. P., A characterization of tangential regularity, Nonlinear Anal., Theory, Meth. Appl. 5 (1981), 625–663.Google Scholar
15. Penot, J. P., A characterization of Clarke's strict tangent cone via nonlinear semigroup, (to appear).Google Scholar
16. Penot, J. P., The Clarke's tangent cone and limits of tangent cones, (to appear).Google Scholar
17. Rockafellar, R. T., Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math, of Operations Research 6 (1981), 424–436.Google Scholar
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).Google Scholar