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A Survey of Clarke’s Subdifferential and the Differentiability of Locally Lipschitz Functions

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Progress in Optimization

Part of the book series: Applied Optimization ((APOP,volume 30))

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Abstract

We survey the theory developed around the Clarke subdifferential to study the variety of differentiability properties of real-valued locally Lipschitz functions on Banach spaces.

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References

  1. J.M. Borwein, Minimal cuscos and subgradients of Lipschitz functions. In Fixed Point Theory and its Applications, J-B Baillon and M. Thera, eds.. Pitman Lecture Notes in Mathematics, Longman 1991, 57–82.

    Google Scholar 

  2. J.M. Borwein and S.P. Fitzpatrick, Characterisation of Clarke subgradients among one-dimensional multifunctions. The Proceedings of Optimization Miniconference II, B.M. Glover and V. Jeyakumer eds, University of Ballarat, 1995, 61–73.

    Google Scholar 

  3. J.M. Borwein, S.P. Fitzpatrick and J.R. Giles, The differentiability of real functions on normed linear spaces using generalized subgradients. Journal of Mathematical Analysis and Applications, 128, 1987, 512–534.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.M. Borwein, S.P. Fitzpatrick and Petar Kenderov, Minimal convex uscos and monotone operators on small sets. Canadian Journal of Mathematics, 43, 1991, 461–476.

    Article  MathSciNet  MATH  Google Scholar 

  5. J.M. Borwein and W.B. Moors, Null sets and essentially smooth Lipschitz functions. SIAM Journal on Optimization, 8, 1998, 309–323.

    Article  MathSciNet  MATH  Google Scholar 

  6. J.M. Borwein and W.B. Moors, Essentially smooth Lipschitz functions. Journal of Functional Analysis, 49, 1997, 305–351.

    Article  MathSciNet  Google Scholar 

  7. J.M. Borwein and W.B. Moors, Lipschitz functions with minimal Clarke subdifferential mappings. The Proceedings of Optimization Miniconference III, B.M. Glover, B.D. Craven and D. Ralph eds, University of Ballarat, 1996, 5–11.

    Google Scholar 

  8. J.M. Borwein and W.B. Moors, Separable determination of integrability and minimality of the Clarke subdifferential mapping. To appear in Proceedings of the American Mathematical Society.

    Google Scholar 

  9. J.M. Borwein and W.B. Moors, A chain rule for essentially smooth Lipschitz functions. SIAM Journal on Optimization, 8, 1998, 300–308.

    Article  MathSciNet  MATH  Google Scholar 

  10. J.M. Borwein and X. Wang, Distinct differentiate functions may share the same Clarke subgradient at all points. Proceedings of the American Mathematical Society, 125, 807–813.

    Google Scholar 

  11. Jens Peter Reus Christensen, Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings, II. Coll. Anal. Fnl.Bordeaux, 1973, 29–39.

    Google Scholar 

  12. F.H. Clarke, Generalized gradients and applications. Transactions of American Mathematical Society, 205, 1975, 247–262.

    Article  MATH  Google Scholar 

  13. F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley Interscience, 1983.

    MATH  Google Scholar 

  14. M. M. Cŏban, P. S. Kenderov and J. P. Revalski, Densely defined selections of multivalued mappings. Transactions of American Mathematical Society, 344, 1994, 533–552.

    Article  MATH  Google Scholar 

  15. R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, 64, Longman, 1993.

    Google Scholar 

  16. S. Fitzpatrick, Metric projections and the differentiability of distance functions. Bulletin of the Australian Mathematical Society, 22, 1980, 291–312.

    Article  MathSciNet  MATH  Google Scholar 

  17. J.R. Giles, On the characterisation of Asplund spaces. Journal of the Australian Mathematical Society, 32A, 1982, 134–144.

    Article  MathSciNet  MATH  Google Scholar 

  18. J.R. Giles, Generic differentiability of locally Lipschitz functions on product spaces. Bulletin of the Australian Mathematical Society, 52, 1995, 487–498.

    Article  MathSciNet  MATH  Google Scholar 

  19. J.R. Giles and M.O. Bartlett, Modified continuity and a generalisation of Michael’s selection theorem. Set-Valued Analysis, 1, 1993, 365–378.

    Article  MathSciNet  MATH  Google Scholar 

  20. J.R. Giles and W.B. Moors, Generic continuity of restricted weak upper semi-continuous set-valued mappings. Set-Valued Analysis, 4, 1996, 25–39.

    Article  MathSciNet  MATH  Google Scholar 

  21. J.R. Giles and W.B. Moors, Generically continuous selections and the differentiability of locally Lipschitz functions. In The Proceedings of Optimization Miniconference III, B. M. Glover, B. D. Craven and D. Ralph, eds., University of Ballarat, 1996, 39–46.

    Google Scholar 

  22. J.R. Giles and Scott Sciffer, Continuity characterisations of differentiability of locally Lipschitz functions. Bulletin of the Australian Mathematical Society, 41, 1990, 371–380.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.R. Giles and Scott Sciffer, Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces. Bulletin of the Australian Mathematical Society, 47, 1993, 205–212.

    Article  MathSciNet  MATH  Google Scholar 

  24. J.R. Giles and Scott Sciffer, Generalising generic differentiability properties from convex to locally Lipschitz functions. Journal of Mathematical Analysis and Applications, 188, 1994, 833–854.

    Article  MathSciNet  MATH  Google Scholar 

  25. K-S Lau, Almost Chebychev subsets in reflexive Banach spaces. Indiana University Mathematics Journal, 27, 1978, 791–795.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Lebourg, Valeur moyenne pour gradient généralisée. Comptes Rendus de l’Académie des Sciennces, 281, 1975, 795–797.

    MathSciNet  MATH  Google Scholar 

  27. W.B. Moors, A characterisation of minimal subdifferential mappings of locally Lipschitz functions. Set-Valued Analysis, 3, 1995, 129–141.

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Michel and J-P Penot, Calcul sous-différential pour les fonctions Lips- chitzienne et non-Lipschitzienne. Comptes Rendus de l’Academie des Sciennces, 298, 1984, 269–272.

    MathSciNet  MATH  Google Scholar 

  29. R.R. Phelps, Gaussian null sets and differentiability of Lipschitz functions on Banach spaces. Pacific Journal of Mathematics, 77, 1978, 523–531.

    MathSciNet  MATH  Google Scholar 

  30. R.R. Phelps, Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics, 1364, Springer-Verlag 2nd ed., 1993.

    Google Scholar 

  31. D. Pompeiu, Sur les fonctions dérivées. Mathematische Annalen, 63, 1907, 326–332.

    Article  MathSciNet  MATH  Google Scholar 

  32. D. Preiss, R.R. Phelps and I. Namioka, Smooth Banach spaces, weak Asplund spaces and monotone usco mappings. Israel Journal of Mathematics, 72, 1990, 257–279.

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Preiss, Differentiability of Lipschitz functions on Banach spaces. Journal of Functional Analysis, 91, 1990, 312–345.

    Article  MathSciNet  MATH  Google Scholar 

  34. H. Rademacher, Über partielle und total Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale. Mathematische Annalen, 79, 1919, 340–359.

    Article  MathSciNet  Google Scholar 

  35. R.T. Rockafellar, The theory of subgradients and its applications to problems of optimization. Convex and non-convex functions. Heldermann- Verlag, 1981.

    Google Scholar 

  36. L. Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued functions. Nonlinear Analysis. Theory, Methods and Applications, 6, 1982, 1037–1053.

    Article  MathSciNet  MATH  Google Scholar 

  37. L. Zajicek, Differentiability of the distance function and points of mulit- valuedness of the metric projection in Banach spaces. Czechoslovak Mathematical Journal, 33, 1983, 292–308.

    MathSciNet  Google Scholar 

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© 1999 Kluwer Academic Publishers

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Giles, J.R. (1999). A Survey of Clarke’s Subdifferential and the Differentiability of Locally Lipschitz Functions. In: Progress in Optimization. Applied Optimization, vol 30. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3285-5_1

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  • DOI: https://doi.org/10.1007/978-1-4613-3285-5_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3287-9

  • Online ISBN: 978-1-4613-3285-5

  • eBook Packages: Springer Book Archive

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