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Proximal analysis in smooth spaces

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Abstract

We provide a highly-refined sequential description of the generalized gradients of Clarke and approximate G-subdifferential of a lower semicontinuous extended-real-valued function defined on a Banach space with a β-smooth equivalent renorm. In the case of a Fréchet differentiable renorm, we give a corresponding result for the corresponding singular objects.

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References

  1. BorweinJ. M.: Minimal cuscos and subgradients of Lipschitz functions, in: J.-B.Baillon and M.Thera (eds), Fixed Point Theory and its Applications, Pitman Lecture Notes in Math., Longman, Harlow, 1991, pp. 57–82.

    Google Scholar 

  2. BorweinJ. M. and FitzpatrickS.: A weak Hadamard smooth renorming of L 1 (Ω, μ), Canad. Math. Bull. 36 (1993), 407–413.

    Google Scholar 

  3. Borwein, J. M. and Fitzpatrick, S.: Weak-star sequential compactness and bornological limit derivatives, Convex Analysis, in press.

  4. BorweinJ. M., FitzpatrickS. and KenderovP.: Minimal convex uscos and monotone operators on small sets, Canad. J. Math. 43 (1991), 461–477.

    Google Scholar 

  5. BorweinJ. M. and PreissD.: A smooth variational principle with application to subdifferentiability and differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517–527.

    Google Scholar 

  6. BorweinJ. M. and StrojwasH. M.: Proximal analysis and boundaries of closed sets in Banach space, Part I: Theory, Canad. J. Math. 38 (1986), 431–452.

    Google Scholar 

  7. BorweinJ. M. and StrojwasH. M.: Proximal analysis and boundaries of closed sets in Banach space, Part II: Applications, Canad. J. Math. 39 (1987), 428–472.

    Google Scholar 

  8. ClarkeF.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

    Google Scholar 

  9. Deville, R. and El Haddad, M.: The Subdifferential of the Sum of Two Functions in Banach Spaces 1. First Order Case, Preprint.

  10. DevilleR., GodefroyG., and ZizlerV.: Smoothness and Renormings and Differentiability in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math. 64, Longman, Harlow, 1993.

    Google Scholar 

  11. DiestelJ.: Geometry of Banach Spaces-Selected Topics, Lecture Notes in Math. 485, Springer-Verlag, Berlin, 1975.

    Google Scholar 

  12. DiestelJ.: Sequences and Series in Banach Spaces, Graduate Texts in Math. Springer-Verlag, New York, Berlin, Tokyo, 1984.

    Google Scholar 

  13. EkelandJ.: On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.

    Google Scholar 

  14. Ioffe, A.: Approximate subdifferentials of nonsmooth functions, CEREMADE Univ. Paris-Dauphine, Publication 8120, 1981.

  15. IoffeA.: Calculus of Dini subdifferentials of functions and contingent derivatives of set-valued maps, Nonlinear Anal. 8 (1984), 517–539.

    Google Scholar 

  16. IoffeA.: Approximate subdifferentials and applications 3. The metric theory, Mathematika 36 (1989), 1–38.

    Google Scholar 

  17. IoffeA.: Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175–192.

    Google Scholar 

  18. KrugerA.: Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Siberian J. Math. 26 (1985), 370–379.

    Google Scholar 

  19. KrugerA. and MordukhovichB.: Extremal points and Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24 (1980), 684–687 (in Russian).

    Google Scholar 

  20. LarmanD. and PhelpsR. R.: Gateaux differentiability of convex functions on Banach spaces, J. London. Math. Soc. 20 (1979), 115–127.

    Google Scholar 

  21. LoewenP. D.: Limits of Fréchet normals in nonsmooth analysis, in: A.Ioffe, M.Marcus, and S.Reich (eds), Optimization and Nonlinear Analysis, Longman, Harlow, 1992, pp. 178–188.

    Google Scholar 

  22. Mordukhovich, B. and Shao, Y.: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math Soc. in press.

  23. Mordukhovich, B. S. and Shao, Y.: Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. in press.

  24. PhelpsR. R.: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer-Verlag, New York, Berlin, Tokyo, 1988, 2nd edn, 1993.

    Google Scholar 

  25. PreissD.: Fréchet derivatives of Lipschitzian functions, J. Funct. Anal. 91 (1990), 312–345.

    Google Scholar 

  26. TreimanJ. S.: Clarke's gradients and epsilon-subgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 65–78.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Simon Fraser University, V5A 1S6, Burnaby, BC, Canada

    Jonathan M. Borwein

  2. Department of Mathematics, Technion-Israel Institute of Technology, 32 000, Haifa, Israel

    Alexander Ioffe

Authors
  1. Jonathan M. Borwein
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  2. Alexander Ioffe
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Borwein, J.M., Ioffe, A. Proximal analysis in smooth spaces. Set-Valued Anal 4, 1–24 (1996). https://doi.org/10.1007/BF00419371

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