Abstract

We study a concept of subdifferential for general extended-real-valued functions defined on arbitrary Banach spaces. At any point considered this basic subdifferential is a nonconvex set of subgradients formed by sequential weak-star limits of the so-called Fréchet ε-subgradients of the function at points nearby. It is known that such and related constructions possess full calculus under special geometric assumptions on Banach spaces where they are defined. In this paper we establish some useful calculus rules in the general Banach space setting.

Author information

Contact details are reproduced from the original publication and may be historical.

Boris S. Mordukhovich

Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A.

Yongheng Shao

Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A.

Keywords

  • Nonsmooth analysis
  • generalized differentiation
  • extended-real-valued functions
  • nonconvex calculus
  • Banach spaces
  • sequential limits

References

  1. J. M. Borwein, H. M. Strojwas: Proximal analysis and boundaries of closed sets in Banach spaces. I: Theory, Can. J. Math. 38 (1986), 431–452.https://doi.org/10.4153/CJM-1986-022-4Google Scholar
  2. F. H. Clarke: Optimizatiom and Nonsmooth Analysis, Wiley, 1983.Google Scholar
  3. F. H. Clarke: Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 57, Soc. Indust. Appl. Math., Philadelphia, Pa., 1989.Google Scholar
  4. A. D. Ioffe: Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 389–416.https://doi.org/10.1090/S0002-9947-1984-0719677-1Google Scholar
  5. A. D. Ioffe: Approximate subdifferential and applications. III: The metric theory, Mathematika 36 (1989), 1–38.https://doi.org/10.1112/S0025579300013541Google Scholar
  6. A. D. Ioffe: Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175–192.https://doi.org/10.1112/jlms/s2-41.1.175Google Scholar
  7. A. Jourani, L. Thibault: A note on Fréchet and approximate subdifferentials of composite functions, Bull. Austral. Math. Soc. 49 (1994), 111–116.https://doi.org/10.1017/S0004972700016142Google Scholar
  8. A. Jourani, L. Thibault: Extensions of subdifferential calculus rules in Banach spaces and applications, Can. J. Math. (to appear)Google Scholar
  9. A. Y. Kruger: Properties of generalized differentials, Siberian Math. J. 26 (1985), 822–832.https://doi.org/10.1007/BF00969103Google Scholar
  10. A. Y. Kruger, B. S. Mordukhovich: Extremal points and Euler equations in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24 (1980), 684–687.Google Scholar
  11. A. Y. Kruger, B. S. Mordukhovich: Generalized normals and derivatives, and necessary optimality conditions in nondifferentiable programming, Depon. VINITI # 408-80, 494-80, Moscow, 1980.Google Scholar
  12. E. B. Leach: A note on inverse function theorem, Proc. Amer. Math. Soc. 12 (1961), 694–697.https://doi.org/10.1090/S0002-9939-1961-0126146-9Google Scholar
  13. P. D. Loewen: Limits of Fréchet normals in nonsmooth analysis, Optimization and Nonlinear Analysis (A. D. Ioffe, L. Marcus, and S. Reich, eds.), Pitman Research Notes Math. Ser. 244, 1992, pp. 178–188.Google Scholar
  14. P. D. Loewen: Optimal Control via Nonsmooth Analysis, CRM Proceedings and Lecture Notes, Vol. 2, Amer. Math. Soc., Providence, RI, 1993.https://doi.org/10.1090/crmp/002Google Scholar
  15. P. D. Loewen, R. T. Rockafellar: Optimal control for unbounded differential inclusions, SIAM J. Control Optim. 32 (1994), 442–470.https://doi.org/10.1137/S0363012991217494Google Scholar
  16. B. S. Mordukhovich: Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960–969.https://doi.org/10.1016/0021-8928(76)90136-2Google Scholar
  17. B. S. Mordukhovich: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526–530.Google Scholar
  18. B. S. Mordukhovich: Approximation Methods in Problems of Optimization and Control, “Nauka”, Moscow, 1988.Google Scholar
  19. B. S. Mordukhovich: Sensitivity analysis in nonsmooth optimization, Theoretical Aspects of Industrial Design (D. A. Field and V. Komkov, eds.), SIAM Proc. Appl. Math., vol. 58, Soc. Indust. Appl. Math., Philadelphia, Pa., 1992, pp. 32–46.Google Scholar
  20. B. S. Mordukhovich: Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Trans. Amer. Math. Soc. 343 (1994), 609–658.https://doi.org/10.2307/2154734Google Scholar
  21. B. S. Mordukhovich: Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions, SIAM J. Control Optim. 33 (1995), 882–915.https://doi.org/10.1137/S0363012993245665Google Scholar
  22. B. S. Mordukhovich, Y. Shao: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. (to appear).Google Scholar
  23. J. J. Moreau: Fonctionelles Convexes, Collége de France, Paris, 1966.Google Scholar
  24. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd edition, Lecture Notes in Mathematics, Vol. 1364, Springer, 1993.Google Scholar
  25. R. T. Rockafellar: Convex Analysis, Princeton Univ. Press, Princeton, N.J., 1970.https://doi.org/10.1515/9781400873173Google Scholar
  26. R. T. Rockafellar: Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331-355.https://doi.org/10.1112/plms/s3-39.2.331Google Scholar
  27. R. T. Rockafellar: The Theory of Subgradients and its Applications to Problems of Optimization. Convex and Nonconvex Functions, Heldermann Verlag, Berlin, 1981.Google Scholar
  28. R. T. Rockafellar: Extensions of subgradient calculus with applications to optimization, Nonlinear Anal. 9 (1985), 665–698.https://doi.org/10.1016/0362-546X(85)90012-4Google Scholar
  29. R. T. Rockafellar, R. J-B Wets: Variational Analysis, Springer (to appear).Google Scholar
  30. L. Thibault: On subdifferentials of optimal value functions, SIAM J. Control Optim. 29 (1991), 1019–1036.https://doi.org/10.1137/0329056Google Scholar
  31. J. S. Treiman: Clarke’s gradients and epsilon-subgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 65–78.https://doi.org/10.1090/S0002-9947-1986-0819935-8Google Scholar

Suggested citation

B. S. Mordukhovich, Y. Shao. “On Nonconvex Subdifferential Calculus in Banach Spaces.” Journal of Convex Analysis 2 (1995), No. 1&2, 211–227. https://doi.org/10.68381/jca02014

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1995. Rights now held by Banach Press LLC.