Links between directional derivatives through multidirectional mean value inequalities

  • Authors:

  • Rafael Correa;Pedro Gajardo;Lionel Thibault

  • Affiliations:

  • Universidad de Chile, Centro de Modelamiento Matemático (CNRS UMI 2807), Departamento de Ingeniería Matemática, Casilla 170/3, Correo 3, Santiago, Chile;Universidad Técnica Federico Santa Marí a, Departamento de Matemática, Avda España 1680, Casilla 110-v, Valparaiso, Chile;Université de Montpellier II, Département des Sciences Mathématiques, Place Eugène Bataillon, 34095, Montpellier Cedex 5, France

  • Venue:
  • Mathematical Programming: Series A and B - Nonlinear convex optimization and variational inequalities
  • Year:
  • 2008

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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)].