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Tangent cones and Dini derivatives

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Abstract

We prove a property of the Bouligand tangent cone to the epigraph (or to the graph) of a locally Lipschitz function. It is also shown how this result can be used in determining Dini sequences. Finally, some relationships between such a cone and Dini derivatives are provided.

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References

  1. Bouligand, G.,Sur l'Existence des Demitangentes à une Courbe de Jordan, Fondamenta Mathematicae, Vol. 15, pp. 215–218, 1930.

    Google Scholar 

  2. Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley, New York, New York, 1983.

    Google Scholar 

  3. Rockafellar, R. T.,Clarke's Tangent Cones and the Boundaries of Closed Sets in R n, Nonlinear Analysis: Theory, Methods and Applications, Vol. 3, pp. 145–154, 1979.

    Google Scholar 

  4. Dubovitskij, A. J., andMilyutin, A. A.,Extremum Problems under Constraints, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, Vol. 5, pp. 395–453, 1965 (in Russian).

    Google Scholar 

  5. Aubin, J. P., andEkeland, I.,Applied Nonlinear Analysis, Wiley, New York, New York, 1984.

    Google Scholar 

  6. Aubin, J. P., andCellina, A.,Differential Inclusions, Springer-Verlag, Berlin, Germany, 1984.

    Google Scholar 

  7. Hiriart-Urruty, J. B.,Tangent Cones, Generalized Gradients, and Mathematical Programming in Banach Space, Mathematics of Operations Research, Vol. 4, pp. 79–97, 1979.

    Google Scholar 

  8. Giannessi, F.,Theorems of the Alternative and Optimality Conditions, Journal of Optimization Theory and Applications, Vol. 42, pp. 331–365, 1984.

    Google Scholar 

  9. Ferrero, O.,Some Properties of Generalized Subdifferentials (in press).

  10. Giannessi, F.,Semidifferentiable Functions and Necessary Optimality Conditions, Journal of Optimization Theory and Applications, Vol. 60, pp. 191–243, 1989.

    Google Scholar 

  11. Giannessi, F., Pappalardo, M., andPellegrini, L.,Necessary Optimality Conditions via Image Problem, Nonsmooth Optimization and Related Topics, Edited by F. H. Clarke, V. Demyanov, and F. Giannessi, Plenum Press, New York, New York, pp. 185–217, 1989.

    Google Scholar 

  12. Elster, K. H., andThierfelder, J.,Abstract Cone Approximations and Generalized Differentiability in Nonsmooth Optimization, Optimization, Vol. 19, pp. 315–341, 1988.

    Google Scholar 

  13. Ioffe, A. D.,Calculus of Dini Subdifferentials of Functions and Contingent Coderivatives of Set-Valued Maps, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 8, pp. 517–539, 1984.

    Google Scholar 

  14. Vlach, M.,Approximation Operators in Optimization Theory, Zeitschrift fur Operations Research, Vol. 37, pp. 251–256, 1980.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    M. Pappalardo (Assistant Professor)

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  1. M. Pappalardo
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Communicated by F. Giannessi

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Pappalardo, M. Tangent cones and Dini derivatives. J Optim Theory Appl 70, 97–107 (1991). https://doi.org/10.1007/BF00940506

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