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Some Results on Subanalytic Variational Inclusions

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Handbook of Optimization

Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 38))

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Abstract

This chapter deals with variational inclusions of the form 0 ∈ f (x) + g(x) + F(x) where f is a locally Lipschitz and subanalytic function, g is a Lipschitz function, F is a set-valued map, acting all in ℝn and n is a positive integer. The study of the previous variational inclusion depends on the properties of the function g. The behaviour as been examinated in different cases : when g is the null function, when g possesses divided differences and when g is not smooth and semismooth. We recall and give a summary of some known methods and the last section is very original and is unpublished. In this last section we combine a Newton type method (applied to f) with a secant type method (applied to g) and we obtain superlinear convergence to a solution of the variational inclusion. Our study in the present chapter is in the context of subanalytic functions, which are semismooth functions and the usual concept of derivative is replaced here by the the concept of Clarke’s Jacobian.

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Authors and Affiliations

  1. Département de Mathématiques et Informatique, Laboratoire L.A.M.I.A - EA 4540, Université des Antilles et de la Guyane, Campus de Fouillole, F–97159, Pointe–à–Pitre, France

    Catherine Cabuzel & Alain Pietrus

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  1. Catherine Cabuzel
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  2. Alain Pietrus
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Correspondence to Catherine Cabuzel .

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Editors and Affiliations

  1. Technical University of Ostrava, 17. listopadu 15, Ostrava-Porub, 708 33, Czech Republic

    Ivan Zelinka

  2. , Faculty of Electrical Engineering and, Technical University of Ostrava, 17. listopadu 15, Ostrava-Poruba, 708 33, Czech Republic

    Václav Snášel

  3. Auburn, 98071, USA

    Ajith Abraham

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Cabuzel, C., Pietrus, A. (2013). Some Results on Subanalytic Variational Inclusions. In: Zelinka, I., Snášel, V., Abraham, A. (eds) Handbook of Optimization. Intelligent Systems Reference Library, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30504-7_3

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  • DOI: https://doi.org/10.1007/978-3-642-30504-7_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30503-0

  • Online ISBN: 978-3-642-30504-7

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