Approximate subgradients and coderivatives in R n


We show that in two dimensions or higher, the Mordukhovich-Ioffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Fréchet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be nonconvex almost everywhere for Fréchet differentiable vector-valued Lipschitz functions. Finally we show that for vector-valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be almost everywhere disconnected.

This is a preview of subscription content, log in to check access.


  1. 1.

    Aubin, J. P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston, 1990.

    Google Scholar 

  2. 2.

    Borwein, J. M.: Minimal CUSCOS and subgradients of Lipschitz functions, in: Fixed Point Theory and Its Applications, Pitman Research Notes 252, Longman, Harlow, 1991, pp. 57–81.

    Google Scholar 

  3. 3.

    Borwein J. M. and Fitzpatrick, S. P.: Characterization of Clarke subdifferentials among one-dimensional multifunctions, CECM Preprint 94-006 (1994).

  4. 4.

    Borwein, D., Borwein, J. M. and Wang, X.: Approximate subgradients and coderivatives in R n, CECM Preprint 96-058 (1996).

  5. 5.

    Bruckner, A. M.: Differentiation of Real Functions, Lecture Notes in Math., edited by A. Dold and B. Eckmann, Springer-Verlag, New York, 1978.

    Google Scholar 

  6. 6.

    Burrill, C. W. and Knudsen, J. R.: Real Variables, Holt, Rinehart and Wintson, New York, 1969.

    Google Scholar 

  7. 7.

    Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983.

    Google Scholar 

  8. 8.

    Hiriart-Urruty, J. and Lemarechal, C.: Convex Analysis and Minimization Algorithms I, Springer-Verlag, Berlin, Heidelberg, 1993.

    Google Scholar 

  9. 9.

    Ioffe, A. D.: Approximate subdifferentials and applications 3: The metric theory, Mathematika 36(71) (1989), 1–38.

    Google Scholar 

  10. 10.

    Ioffe, A. D.: Approximate subdifferentials and applications I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 390–416.

    Google Scholar 

  11. 11.

    Ioffe, A. D.: Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Anal. Theory, Methods Appl. 8 (1984), 517–539.

    Google Scholar 

  12. 12.

    Katriel, G.: Are the approximate and the Clarke subgradients generically equal?, J. Math. Anal. Appl 193 (1995), 588–593.

    Google Scholar 

  13. 13.

    Lay, S. R.: Convex Sets and Their Applications, Wiley, New York, 1982.

    Google Scholar 

  14. 14.

    Malý, J.: Darboux property of gradients, Real Analysis Exchange (to appear).

  15. 15.

    Mordukhovich, B. S.: Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960–969.

    Google Scholar 

  16. 16.

    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 

  17. 17.

    Mordukhovich, B. S.: Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988.

    Google Scholar 

  18. 18.

    Mordukhovich, B. S.: Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183 (1994), 250–288.

    Google Scholar 

  19. 19.

    Royden, H. L.: Real Analysis, Macmillan, New York, 1988.

    Google Scholar 

  20. 20.

    Rockafellar, R. T.: Proximal subgradients, marginal values, and augmented lagrangians in nonconvex optimization, Math. Oper. Res. 6(3) (1981), 424–436.

    Google Scholar 

  21. 21.

    Rockafellar, R. T.: Dualization of subgradient conditions for optimality, Gruppo di Ottimizzazione e Ricerca Operativa 3.182 (620), Gennaio, 1992.

  22. 22.

    Stromberg, K. R.: An Introduction to Classical Real Analysis, Wadsworth Internat. Math. Series, 1981.

  23. 23.

    Warga, J.: Fat homeomorphisms and unbounded derivate container, J. Math. Anal. Appl. 81 (1981), 545–560.

    Google Scholar 

Download references

Author information



Additional information

Research supported by NSERC and the Shrum Endowment at Simon Fraser University.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Borwein, D., Borwein, J.M. & Wang, X. Approximate subgradients and coderivatives in R n . Set-Valued Anal 4, 375–398 (1996).

Download citation

Mathematics Subject Classifications (1991)

  • Primary 49J52
  • Secondary 26A27
  • 26B12
  • 49J50
  • 52A20

Key words

  • subgradient
  • coderivative
  • generalized Jacobian
  • Lipschitz function
  • bump function
  • gauge
  • nowhere dense set
  • Lebesgue measure
  • disconnectedness