Set-Valued Analysis

, Volume 6, Issue 2, pp 171–185

The Equivalence of Several Basic Theorems for Subdifferentials

  • Qiji J. Zhu


Several different basic tools are used for studying subdifferentials. They are a nonlocal fuzzy sum rule in (Borwein et al., 1996; Zhu, 1996), a multidirectional mean value theorem in (Clarke and Ledyaev, 1994; Clarke et al., 1998), local fuzzy sum rules in (Ioffe, 1984, 1990) and an extremal principle in (Kruger and Mordukhovich, 1980; Mordukhovich, 1976, 1980, 1994). We show that all these basic results are equivalent and discuss some interesting consequences of this equivalence.

subdifferentials mean value inequalities local fuzzy sum rules nonlocal fuzzy sum rules extremal principles and Asplund spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asplund, E.: Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31–47.Google Scholar
  2. 2.
    Borwein, J. M. and Ioffe, A. D.: Proximal analysis in smooth spaces, CECM Research Report 93–04, 1993, Set-Valued Analysis 4 (1996), 1-24.Google Scholar
  3. 3.
    Borwein, J. M., Mordukhovich, B. S. and Shao, Y.: On the equivalence of some basic principles in variantional analysis, CECM Research Report 97–098, 1997.Google Scholar
  4. 4.
    Borwein, J. M. and Preiss, D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517–527.Google Scholar
  5. 5.
    Borwein, J. M., Treiman, J. S. and Zhu, Q. J.: Partially smooth variational principles and applications, CECM Research Report 96–088, 1996.Google Scholar
  6. 6.
    Borwein, J. M. and Zhu, Q. J.: Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, CECM Research Report 94–12, 1994, SIAM J. Control Optim. 34 (1996), 1586-1591.Google Scholar
  7. 7.
    Clarke, F. H.: Lecture Notes in Nonsmooth Analysis, unpublished.Google Scholar
  8. 8.
    Clarke, F. H. and Ledyaev, Yu. S.: Mean value inequalities in Hilbert space, Trans. Amer. Math. Soc. 344 (1994), 307–324.Google Scholar
  9. 9.
    Clarke, F. H., Ledyaev, Yu. S., Stern, R. J. and Wolenski, P. R.: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, Vol. 178, Springer-Verlag, New York, 1998.Google Scholar
  10. 10.
    Deville, R., Godefroy, G. and Zizler, V.: Smoothness and Renormings and Differentiability in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, No. 64, Wiley, New York, 1993.Google Scholar
  11. 11.
    Deville, R. and Haddad, E. M. E.: The subdifferential of the sum of two functions in Banach spaces, I. First order case, Convex Analysis 3 (1996), 295–308.Google Scholar
  12. 12.
    Deville, R. and Ivanov, M.: Smooth variational principles with constraints, Arch. Math. (Basel) 69 (1997), 418–426.Google Scholar
  13. 13.
    Ekeland, I.: Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443–474.Google Scholar
  14. 14.
    Fabian, M.: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolinae 30 (1989), 51–56.Google Scholar
  15. 15.
    Ioffe, A. D.: On subdifferentiability spaces, Ann. New York Acad. Sci. 410 (1983), 107–112.Google Scholar
  16. 16.
    Ioffe, A. D.: Calulus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Anal. 8 (1984), 517–539.Google Scholar
  17. 17.
    Ioffe, A. D.: Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175–192.Google Scholar
  18. 18.
    Ioffe, A. D. and Rockafellar, T. R.: The Euler and Weierstrass conditions for nonsmooth variational problems, Calc. Var. 4 (1996), 59–87.Google Scholar
  19. 19.
    Kruger, A. Y. and Mordukhovich, B. S.: Extremal points and Euler equations in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24 (1980), 684–687.Google Scholar
  20. 20.
    Mordukhovich, B. S.: Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960–969.Google Scholar
  21. 21.
    Mordukhovich, B. S.: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526–530.Google Scholar
  22. 22.
    Mordukhovich, B. S.: Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183 (1994), 250–288.Google Scholar
  23. 23.
    Mordukhovich, B. S. and Shao, Y.: Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124 (1996), 197–205.Google Scholar
  24. 24.
    Mordukhovich, B. S. and Shao, Y.: Nonconvex differential calculus for infinite dimensional multifunctions, Set-Valued Analysis 4 (1996), 205–236.Google Scholar
  25. 25.
    Mordukhovich, B. S. and Shao, Y.: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), 1235–1280.Google Scholar
  26. 26.
    Rockafellar, R. T.: Proximal subgradients, marginal values, and augmented Lagragians in nonconvex optimization, Math. Oper. Res. 6 (1981), 424–436.Google Scholar
  27. 27.
    Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics 1364, Springer-Verlag, New York, Berlin, Tokyo, 1988, Second Edition 1993.Google Scholar
  28. 28.
    Vanderwerff, J. and Zhu, Q. J.: A limiting example for the local ‘fuzzy’ sum rule in nonsmooth analysis, CECM Research Report 96–083, 1996, to appear in Proc. Amer. Math. Soc. Google Scholar
  29. 29.
    Zhu, Q. J.: Clarke-Ledyaev mean value inequality in smooth Banach spaces, CECM Research Report 96–78, 1996, to appear in Nonlinear Anal. Google Scholar
  30. 30.
    Zhu, Q. J.: Subderivatives and applications, in: Proceedings of International Conference on Dynamical Systems and Differential Equations, Springfield, MO, May 29–June 1, 1996.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Qiji J. Zhu
    • 1
  1. 1.Department of Mathematics and StatisticsWestern Michigan UniversityKalamazooU.S.A.

Personalised recommendations