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Directionally Lipschitzian Mappings on Baire Spaces

  • J. M. Borwein (a1) and H. M. Stròjwas (a2)
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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.

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References
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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.
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.
3. Bouligand, G., Introduction à la géométrie infinitésimale directe (Vuibert, Paris, 1932).
4. Brøndsted, A. and Rockafellar, R. T., On the subdifferentiability of convex functions, Proc. Amer Math. Soc. 16 (1965), 605–11.
5. Clarke, F. H., Necessary conditions for nonsmooth problems in optimal control and the calculus of variation, Thesis, University of Washington, Seattle (1973).
6. Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–62.
7. Clarke, F. H., Generalized gradients of Lipschitz functions, Adv. in Math. 40 (1981), 5267.
8. Clarke, F. H., A new approach to Lagrange multipliers, Math of Operations Research 1 (1976), 165–74.
9. Dolecki, S., Tangency and differentiation: some applications of convergence theory, Annali di Matematica pura ed applicata 130 (1982), 223255.
10. Day, M. M., Normed linear spaces, (Springer-Verlag, New York, 1973).
11. Diestel, J., Geometry of Banach spaces — selected topics (Springer-Verlag, New York, 1975).
12. Hiriart-Urruty, J. B., Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math, of Op. Res. 4 (1975), 7997.
13. Holmes, R. B., Geometric functional analysis (Springer-Verlag, New York, 1975).
14. Klee, V., On a question of Bishop and Phelps, Amer. J. Math. 85 (1963), 9598.
15. Klee, V., Extremal structure of convex sets 11, Math. A. 69 (1958), 90104.
16. Lebourg, G., Valeur moyenne pour gradient généralisé, C. R. Acad. Aci. Paris, Ser. A. 281 (1975), 795797.
17. Lebourg, G., Generic differentiability of Lipschitzian functions, Trans. Am. Math. Soc. 256 (1979), 125144.
18. McLinden, L., An application of Ekeland's theorem to minimax problems, Nonlin. Anal. Th Meth. Appl. 6 (1982), 189196.
19. Namioka, I., Separate continuity and joint continuity, Pacific J. Math 57 (1974), 515531.
20. Peck, N., Support points in locally convex spaces, Duke Math. J. 38, 271278.
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, 495511, Athenaum, Berlin.
22. Penot, J. P., A characterization of tangential regularity, Nonlinear Anal., Theory, Math. Appl. 5 (1981), 625663.
23. Rockafellar, R. T., Directionally Lipschitzian functions and subdifferential calculus, Bull. London, Math. Soc. 39 (1979), 331355.
24. Rockafellar, R. T., Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math 32 (1980), 257280.
25. Rockafellar, R. T., Clarke's tangent cones and the boundaries of closed sets in Rn, Nolin. Analysis. Th. Meth Appl. 3 (1979), 145154.
26. Rockafellar, R. T., Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, to appear in Math. Prog. Study 15.
27. Rockafellar, R. T., The theory of subgradients and its applications to problems of optimization. Convex and nonconvex functions (Heldermann-Verlag, Berlin, 1981).
28. Wilansky, A., Modern methods in topological vector spaces (McGraw-Hill, New York, 1978)
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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