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Links between directional derivatives through multidirectional mean value inequalities

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Abstract

We prove in the general setting of lower semicontinuous functions on Banach spaces the relation between the Rockafellar directional derivative and the mixed lower limit of the lower Dini derivatives. As a byproduct we derive the famous inclusions of tangent cones of closed sets in Banach spaces. The results are established using as principal tool multidirectional mean value inequalities [Aussel et al., SIAM J Optim 9(3), 690–706 (1999)].

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Chile, Centro de Modelamiento Matemático (CNRS UMI 2807), Casilla 170/3, Correo 3, Santiago, Chile

    Rafael Correa

  2. Departamento de Matemática, Universidad Técnica Federico Santa Marí a, Avda España 1680, Casilla 110-v, Valparaiso, Chile

    Pedro Gajardo

  3. Département des Sciences Mathématiques, Université de Montpellier II, Place Eugène Bataillon, 34095, Montpellier Cedex 5, France

    Lionel Thibault

Authors
  1. Rafael Correa
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  2. Pedro Gajardo
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  3. Lionel Thibault
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Corresponding author

Correspondence to Pedro Gajardo.

Additional information

To Alfred Auslender, on the occasion of his 65th birthday and of his award of a Dhc.

This research was partially supported by Ecos-Conicyt project C04E03.

The second author was partially supported by FONDAP-Conicyt en Matemáticas Aplicadas, Centro de Modelamiento Matemático, Universidad de Chile (CNRS UMI 2807) and Proyect Fondecyt N 3060068.

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Correa, R., Gajardo, P. & Thibault, L. Links between directional derivatives through multidirectional mean value inequalities. Math. Program. 116, 57–77 (2009). https://doi.org/10.1007/s10107-007-0131-7

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