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Approximation and regularisation of arbitrary sets in finite dimension

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Abstract

We define the regularisation of a set by using two kernels. This procedure generalizes that defined by Laory-Lions for the functions. We privilege the geometrical point of view and we obtain in particular some new results about the asymptotic behaviour of Clarke's normal cone at a point of a set belonging to the regularized family

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References

  1. Attouch, H.:Variational convergences for Functions and Operators, Applicable Mathematics Series, Pitman London, 1984.

    Google Scholar 

  2. Attouch, H. and Azé, D.: Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions methods, to appear.

  3. Attouch, H. and Wets, R. J.-P.: in H. Attouch, J.-P. Aubin, F. H. Clarke, and I. Ekeland (eds.),Epigraphical Analysis, Analyse non linéaire, Gauthie-Villars Paris et C.R.M., Montreal, 1989, pp. 73–100.

    Google Scholar 

  4. Benoist, J.: Approximation and Regularization of arbitrary sets: the finite dimension case, Publications du département de mathématiques de Limoges (1993).

  5. Bougeard, M., Penot J.-P., and Pommelet, A.: Towards Minimal Assumptions for the Infimal Convolution Regularization,J. Approx. Theory 64 (3) (1991), 245–270.

    Google Scholar 

  6. Brezis, H.:Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.

    Google Scholar 

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

    Google Scholar 

  8. Debreu, G.:Theory of value, Wiley, New York, 1959.

    Google Scholar 

  9. Hiriart-Urruty, J.-B.: How to regularize a difference of convex functions,J. Math. Anal. Appl. 162 (1991), 196–209.

    Google Scholar 

  10. Ioffe, A. D.: Approximate subdifferentials and applications I: the finite dimensional theory,Trans. Am. Math. Soc. 281 (1984), 389–416.

    Google Scholar 

  11. Lasry J.-M. and Lions, P.-L.: A remark on regularization in Hilbert spaces,Israel J. Math. 55 (1986), 257–266.

    Google Scholar 

  12. Mordukhovich, B. Sh.: Maximum principle in the optimal time control problem with non-smooth constraints,Prikl. Mat. Meh. 40 (1976), 1014–1023.

    Google Scholar 

  13. Moreau, J.-J.: Fonctionnelles convexes, Lecture notes, Collège de France, Paris, 1967.

    Google Scholar 

  14. Poliquin, R.: An extension of Attouch's theorem and its application to second-order epidifferentiation of convexly composite functions, to appear.

  15. Rockafellar, R. T.:Convex Analysis, Princeton University Press, 1970.

  16. Rockafellar, R. T.: Extensions of subgradient calculus with applications to optimization,Nonlinear Anal. 9 (1985), 665–698.

    Google Scholar 

  17. Volle, M.: Régularisation des fonctions fortement minorées dans les espaces de Hilbert, Séminaire d'Analyse convexe, Montpellier, 1990.

  18. Yosida, K.:Functional Analysis, third edition, Springer, Berlin, Heidelberg, New-York, 1971.

    Google Scholar 

  19. Zolezzi, T.: Continuity of generalized gradients and multipliers under perturbations,Math. Oper. Res. 10 (1985), 664–673.

    Google Scholar 

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Benoist, J. Approximation and regularisation of arbitrary sets in finite dimension. Set-Valued Anal 2, 95–115 (1994). https://doi.org/10.1007/BF01027095

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