Hostname: page-component-cc8bf7c57-j4qg9 Total loading time: 0 Render date: 2024-12-11T22:16:22.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Directionally Lipschitzian Mappings on Baire Spaces

Published online by Cambridge University Press:  20 November 2018

J. M. Borwein  and
H. M. Stròjwas
J. M. Borwein
Affiliation:
Dalhousie University, Halifax, Nova Scotia
H. M. Stròjwas
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.

Studies of optimization problems have led in recent years to definitions of several types of generalized directional derivatives. Those derivatives of primary interest in this paper were introduced and investigated by F. M. Clarke ([5], [6], [7], [8]), J. B. Hiriart-Urruty ([12]), Lebourg ([16], [17]), R. T. Rockafellar ([23], [24], [26], [27]), Penot ([21], [22]) among others.

In an attempt to explore in more detail relationships between various types of generalized directional derivatives we discovered some unexpected results which were not observed by the above mentioned authors. We are able to give simple conditions which characterize directionally Lipschitzian functions defined on a Baire metrizable locally convex topological vector space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 1 , 01 February 1984 , pp. 95 - 130
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Borwein, J. M., Stability and regular points of inequality systems, Research Report 82–2, Department of Mathematics, Mellon Institute of Science, Carnegie-Mellon University. To appear in Numerical Functional Analysis and Applications.CrossRefGoogle Scholar
2. Borwein, J. M. and Stròjwas, H. M., Directionally Lipschitzian mappings on Baire spaces, Research Report 83–3, Department of Mathematics, Mellon Institute of Science, Carnegie-Melon University.CrossRefGoogle Scholar
3. Bouligand, G., Introduction à la géométrie infinitésimale directe (Vuibert, Paris, 1932).Google Scholar
4. Brøndsted, A. and Rockafellar, R. T., On the subdifferentiability of convex functions, Proc. Amer Math. Soc. 16 (1965), 605–11.Google Scholar
5. Clarke, F. H., Necessary conditions for nonsmooth problems in optimal control and the calculus of variation, Thesis, University of Washington, Seattle (1973).CrossRefGoogle Scholar
6. Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–62.Google Scholar
7. Clarke, F. H., Generalized gradients of Lipschitz functions, Adv. in Math. 40 (1981), 52–67.Google Scholar
8. Clarke, F. H., A new approach to Lagrange multipliers, Math of Operations Research 1 (1976), 165–74.Google Scholar
9. Dolecki, S., Tangency and differentiation: some applications of convergence theory, Annali di Matematica pura ed applicata 130 (1982), 223–255.Google Scholar
10. Day, M. M., Normed linear spaces, (Springer-Verlag, New York, 1973).CrossRefGoogle Scholar
11. Diestel, J., Geometry of Banach spaces — selected topics (Springer-Verlag, New York, 1975).CrossRefGoogle Scholar
12. Hiriart-Urruty, J. B., Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math, of Op. Res. 4 (1975), 79–97.Google Scholar
13. Holmes, R. B., Geometric functional analysis (Springer-Verlag, New York, 1975).Google Scholar
14. Klee, V., On a question of Bishop and Phelps, Amer. J. Math. 85 (1963), 95–98.Google Scholar
15. Klee, V., Extremal structure of convex sets 11, Math. A. 69 (1958), 90–104.Google Scholar
16. Lebourg, G., Valeur moyenne pour gradient généralisé, C. R. Acad. Aci. Paris, Ser. A. 281 (1975), 795–797.Google Scholar
17. Lebourg, G., Generic differentiability of Lipschitzian functions, Trans. Am. Math. Soc. 256 (1979), 125–144.Google Scholar
18. McLinden, L., An application of Ekeland's theorem to minimax problems, Nonlin. Anal. Th Meth. Appl. 6 (1982), 189–196.Google Scholar
19. Namioka, I., Separate continuity and joint continuity, Pacific J. Math 57 (1974), 515–531.Google Scholar
20. Peck, N., Support points in locally convex spaces, Duke Math. J. 38, 271–278.Google Scholar
21. Penot, J. P., The use of generalized subdifferential calculus in optimization theory, Proc. Third Symp. on Operations Research, Mannheim (1978), Methods of Operations Research 31, 495–511, Athenaum, Berlin.Google Scholar
22. Penot, J. P., A characterization of tangential regularity, Nonlinear Anal., Theory, Math. Appl. 5 (1981), 625–663.Google Scholar
23. Rockafellar, R. T., Directionally Lipschitzian functions and subdifferential calculus, Bull. London, Math. Soc. 39 (1979), 331–355.Google Scholar
24. Rockafellar, R. T., Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math 32 (1980), 257–280.Google Scholar
25. Rockafellar, R. T., Clarke's tangent cones and the boundaries of closed sets in Rn, Nolin. Analysis. Th. Meth Appl. 3 (1979), 145–154.Google Scholar
26. Rockafellar, R. T., Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, to appear in Math. Prog. Study 15.CrossRefGoogle Scholar
27. Rockafellar, R. T., The theory of subgradients and its applications to problems of optimization. Convex and nonconvex functions (Heldermann-Verlag, Berlin, 1981).Google Scholar
28. Wilansky, A., Modern methods in topological vector spaces (McGraw-Hill, New York, 1978)Google Scholar