Set-Valued Analysis

, Volume 4, Issue 4, pp 375–398 | Cite as

Approximate subgradients and coderivatives in R n

  • D. Borwein
  • J. M. Borwein
  • Xianfu Wang


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.

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 


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  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).Google Scholar
  4. 4.
    Borwein, D., Borwein, J. M. and Wang, X.: Approximate subgradients and coderivatives in R n, CECM Preprint 96-058 (1996).Google Scholar
  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).Google Scholar
  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.Google Scholar
  22. 22.
    Stromberg, K. R.: An Introduction to Classical Real Analysis, Wadsworth Internat. Math. Series, 1981.Google Scholar
  23. 23.
    Warga, J.: Fat homeomorphisms and unbounded derivate container, J. Math. Anal. Appl. 81 (1981), 545–560.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • D. Borwein
    • 1
  • J. M. Borwein
    • 2
  • Xianfu Wang
    • 3
  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada
  2. 2.Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada
  3. 3.Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada

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