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

Borwein, J. M. ; Strojwas, H. M.

Canadian journal of mathematics, Tome 38 (1986) no. 2, pp. 431-452

Résumé

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.

Détail

DOI : 10.4153/CJM-1986-022-4

Bibliographie

[1] 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] 2. Borwein, J. M., Tangent cones, starshape and convexity, Internat. J. Math. & Math. Sci. 1 (1978), 459–477. Google Scholar

[3] 3. Borwein, J. M., Weak local sup port ability and applications to approximation, Pacific Journal of Mathematics 82 (1979), 323–338. Google Scholar

[4] 4. Borwein, J. M. and O'Brien, R., Tangent cones and convexity, Canadian Math. Bull. 19 (1976), 257–261. Google Scholar

[5] 5. Borwein, J. M. and Strojwas, H., Tangential approximations, to appear in Nonlinear Anal, Theory, Meth. Appl. Google Scholar

[6] 6. Clarke, F. H., Optimization and nonsmooth analysis (John Wiley, 1982). Google Scholar

[7] 7. Cornet, B., Regular properties of tangent and normal cones, (to appear). Google Scholar

[8] 8. Diestel, J., Geometry of Banach spaces — selected topics (Springer-Verlag, 1975). Google Scholar | DOI

[9] 9. Ekeland, I., On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. Google Scholar

[10] 10. Holmes, R. B., Geometric functional analysis and its applications (Springer-Verlag, 1975). Google Scholar | DOI

[11] 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] 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] 13. Lau, K., Almost Chebyshev subsets in reflexive Banach spaces, Indianna Univ. Math. J. 27 (1978), 791–795. Google Scholar

[14] 14. Penot, J. P., A characterization of tangential regularity, Nonlinear Anal., Theory, Meth. Appl. 5 (1981), 625–663. Google Scholar

[15] 15. Penot, J. P., A characterization of Clarke's strict tangent cone via nonlinear semigroup, (to appear). Google Scholar

[16] 16. Penot, J. P., The Clarke's tangent cone and limits of tangent cones, (to appear). Google Scholar

[17] 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] 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