Advertisement

Set-Valued Analysis

, Volume 11, Issue 1, pp 37–67 | Cite as

Local Lipschitz-constant Functions and Maximal Subdifferentials

  • J. M. Borwein
  • J. Vanderwerff
  • Xianfu Wang
Article

Abstract

It is shown that if k(x) is an upper semicontinuous and quasi lower semicontinuous function on a Banach space X, then k(x)BX* is the Clarke subdifferential of some locally Lipschitz function on X. Related results for approximate subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke subdifferential, one can obtain a maximal Clarke subdifferential map via its ‘local Lipschitz-constant’ function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed.

Lipschitz function Baire category Clarke subdifferential approximate subdifferential local Lipschitz-constant function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benyamini, Y. and Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, Vol. 1, Amer. Math. Soc. Colloq. Publ. 48, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  2. 2.
    Borwein, J. M.: Minimal CUSCOs and subgradients of Lipschitz functions, In: Fixed Point Theory and its Applications, Pitman Res. Notes 252, 1991, pp. 57-81.Google Scholar
  3. 3.
    Borwein, J. M. and Fitzpatrick, S.: Characterization of Clarke subgradients among one-dimensional multifunctions, In: B.M. Glover and V. Jeyakumar (eds), Proc. of the Optimization Miniconference II, 1995, pp. 61-73.Google Scholar
  4. 4.
    Borwein, J. M. and Ioffe, A. D.: Proximal analysis in smooth spaces, Set-Valued Anal. 4 (1996), 1-24.Google Scholar
  5. 5.
    Borwein, J. M. and Moors, W. B.: Null sets and essentially smooth Lipschitz functions, SIAM J. Optim. 8 (1998), 309-323.Google Scholar
  6. 6.
    Borwein, J. M., Moors, W. B. and Wang, X.: Generalized subdifferentials: a Baire categorical approach, Trans. Amer. Math. Soc. 353 (2001), 3875-3893.Google Scholar
  7. 7.
    Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983.Google Scholar
  8. 8.
    Deville, R., Godefroy, G. and Zizler, V.: Smoothness and Renorming in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math. 64, Longman, Harlow, 1993.Google Scholar
  9. 9.
    Fitzpatrick, S.: Differentiability of real-valued functions and continuity of metric projections, Proc. Amer. Math. Soc. 91 (1984), 544-548.Google Scholar
  10. 10.
    Giles, J. R.: Convex Analysis with Application in Differentiation of Convex Functions, Res. Notes Math. 58, 1982.Google Scholar
  11. 11.
    Ioffe, A. D.: Approximate subdifferentials and applications II, Mathematika 33 (1986), 111-128.Google Scholar
  12. 12.
    Ioffe, A. D.: Approximate subdifferentials and applications 3: the metric theory, Mathematika 36 (1989), 1-38.Google Scholar
  13. 13.
    Mordukhovich, B. S. and Shao, Y.: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), 1235-1280.Google Scholar
  14. 14.
    Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math., Springer-Verlag Berlin Heidelberg, 1993.Google Scholar
  15. 15.
    Preiss, D., Phelps, R. R. and Namioka, I.: Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings, Israel J. Math. 72 (1990), 257-279.Google Scholar
  16. 16.
    Preiss, D.: Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), 312-345.Google Scholar
  17. 17.
    Rockafellar, R. T. and Wets, R. J.-B.: Variational Analysis, Springer-Verlag, Berlin, 1998.Google Scholar
  18. 18.
    Stromberg, K. R.: An Introduction to Classical Real Analysis, Wadsworth International Mathematics Series, 1981.Google Scholar
  19. 19.
    Thibault, L.: On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear Anal. 6 (1982), 1037-1053.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • J. M. Borwein
    • 1
  • J. Vanderwerff
    • 2
  • Xianfu Wang
    • 3
  1. 1.Centre for Experimental and Constructive Mathematics, Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of Mathematics and ComputingLa Sierra UniversityRiversideU.S.A
  3. 3.Department of Mathematics and StatisticsOkanagan University CollegeKelownaCanada

Personalised recommendations