Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T01:54:14.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions Of Subdifferential Calculus Rules in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

A. Jourani  and
L. Thibault
A. Jourani
Affiliation:
Université de Bourgogne Laboratoire d'Analyse Appliquée et Optimisation B.R 138 21004 Dijon cedex, France
L. Thibault
Affiliation:
Université Montpellier II Département des Sciences Mathématiques 34095 Montpellier cedex 5 France
Rights & Permissions [Opens in a new window]

Abstract

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.

This paper is devoted to extending formulas for the geometric approximate subdifferential and the Clarke subdifferential of extended-real-valued functions on Banach spaces. The results are strong enough to include completely the finite dimensional setting.

Keywords

49J5246B2058C20subdifferentialepi-Lipschitz-like setscompactly epi-Lipschitzian sets
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 4 , 01 August 1996 , pp. 834 - 848
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Aubin, J.P., Lipschitz behaviour of solutions to convex minimization problems, Math. Oper., Res. 9(1984), 87–111.Google Scholar
2. Borwein, J.M., Epi-Lipschitz-like sets in Banach space: Theorems and Examples, Nonlinear, Anal. 11(1987), 1207–1217.Google Scholar
3. Borwein, J.M. and Strojwas, H.M., Tangential approximations, Nonlinear, Anal. 9(1985), 1347–1366.Google Scholar
4. Clarke, F.H., Necessary conditions for nonsmooth problems in optimal control and the calculus of variations, Thesis, University of Washington, Seattle (1973).Google Scholar
5. Clarke, F.H., A new approach to Lagrange multipliers, Math. Oper., Res. 1(1976), 165–174.Google Scholar
6. Clarke, F.H., Optimization and Nonsmooth Analysis, John Wiley, New York (1983).Google Scholar
7. Clarke, F.H. and Raissi, N., Formules d'intersection en analyse non lisse, Ann. Sci. Math., Quebec, 14(1990), 121–129.Google Scholar
8. Dolecki, S., Tangency and differentiation: some applications of convergence, Theory, Ann. Mat. Pura, Appl. 130(1982), 235–255.Google Scholar
9. El, B. Abdouni and Thibault, L., Lagrange multipliers for Pareto nonsmooth programming problems in Banach spaces,, Optimization 26(1992), 277–285.Google Scholar
10. Ioffe, A.D., Nonsmooth analysis: differential calculus of non-differentiable mappings, Trans. Amer. Math., Soc. 266(1981), 1–56.Google Scholar
11. Ioffe, A.D., Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math., Soc, 281(1984), 389–416.Google Scholar
12. Ioffe, A.D., Approximate subdifferentials and applications II: Functions on locally convex spaces,, Mathematika, 33(1986), 111–128.Google Scholar
13. Ioffe, A.D., Approximate subdifferentials and applications III: The metric theory,, Mathematika, 36(1989), 1–38.Google Scholar
14. Jourani, A. and Thibault, L., Metric regularity for strongly compactly Lipschitzian mappings, Nonlinear Anal., to appear.Google Scholar
15. Jourani, A., The use of metric graphical regularity in approximate subdifferential calculus rules infinite dimensions,, Optimization 21(1990), 509–519.Google Scholar
16. Jourani, A., Approximate subdifferential of composite functions, Bull. Austral. Math., Soc, 47(1993), 1–13.Google Scholar
17. Kelley, J.L., General topology, Springer-Verlag, New York, (1975).Google Scholar
18. Ya, A. Kruger, Properties of generalized differentials, Sib. Math., J. 26(1985), 1822–1832.Google Scholar
19. Ya, A. Kruger and Mordukhovich, B.S., Extreme points and the Euler equation in nondifferentiable optimization problems, Dokl. Akad. Nauk., BSSR, 24(1980), 684–687.Google Scholar
20. Loewen, P.D., Limits ofFrechet normals in nonsmooth analysis, In: Optimization and Nonlinear Analysis (eds. Ioffe, A., Marcus, M. and Reich, S.), Pitman Research Notes in Mathematics Series, Great Britain (1992).Google Scholar
21. Mordukhovich, B.S., Maximum principle in the problem of time optimal control with nonsmooth constraints, J. Appl. Math., Mech., 40(1976), 960–969.Google Scholar
22. Mordukhovich, B.S., Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings, Dokl. Akad. Nauk., BSSR, 28(1984), 976–979.Google Scholar
23. Mordukhovich, B.S., Approximation Methods in Problems of Optimization and Control, Nauka, Moscow (1988).Google Scholar
24. Radstrom, H., An imbedding theorem for spaces of convex sets, Proc Amer. Math., Soc, 3(1952), 165–169.Google Scholar
25. Rockafellar, R.T., Generalized directional derivatives and subgradients of nonconvex functions, Canad. J., Math. 32(1980), 257–280.Google Scholar
26. Mordukhovich, B.S., Directionally Lipschitzian functions and subdifferential calculus, Proc London Math., Soc, 39(1979), 331–355.Google Scholar
27. Mordukhovich, B.S., Extensions of subgradient calculus with applications to optimization, Nonlinear, Anal. 9(1985), 665–698.Google Scholar
28 Thibault, L., Subdifferentials of compactly Lipschitzian vector valued functions, Ann. Mat. Pura Appl. 125(1980), 157-192.Google Scholar
29 Ward, D. E. and Borwein, J. M., Nonsmooth calculus infinite dimensions, SIAM J. Control Optim. 25(1987), 1312-1340.Google Scholar