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A primal–dual operation on sets linked with closed convex relaxation processes

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Abstract

We give some properties and uses of a primal–dual operation on sets that appear in the closed convex relaxation process (Hiriart-Urruty et al. in Rev. Mat. Iberoam. 27(2):449–474, 2011; López and Volle in J. Conv. Anal. 17(3–4):1057–1075, 2010). Applications are provided concerning a class of relaxed minimization problems in the frame of the so called B-regularization theory. Special attention is paid to the case when the initial problem admits optimal solutions under compactness assumptions.

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References

  • Benoist J, Hiriart-Urruty J-B (1996) What is the subdifferential of the closed convex hull of a function. SIAM J Math Anal 27(6):1661–1679

    Article  Google Scholar 

  • Bot RI, Fratean A-R (2010) Looking for appropriate qualification conditions for subdifferential formulae and dual representations for convex risk measures. arXiv:1005.2487v1

  • Correa R, Hantoute A (2010) New formulas for the Fenchel subdifferential of the conjugate function. Set-Val Var Anal 18:405–422

    Article  Google Scholar 

  • Dunford N, Schwartz JT (1988) Linear operators, part 1. Wiley, New York

    Google Scholar 

  • Filipovic D, Kupper M (2007) Monotone and cash-invariant convex functions and hulls. Insur Math Econ 41(1):1–16

    Article  Google Scholar 

  • Hantoute A, López MA, Zalinescu C (2008) Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions. SIAM J Optim 19(2):863–882

    Article  Google Scholar 

  • Hiriart-Urruty JB, López MA, Volle M (2011) The ϵ-strategy in variational analysis: illustration with the closed convex hull of a function. Rev Mat Iberoam 27(2):449–474

    Article  Google Scholar 

  • Laurent PJ (1972) Approximation et optimization. Hermann, Paris

    Google Scholar 

  • López MA, Volle M (2010) A formula for the set of optimal solutions of a relaxed minimization problem. Applications to subdifferential calculus. J Convex Anal 17(3–4):1057–1075

    Google Scholar 

  • Moreau JJ (1966) Fonctionnelles convexes. Collége de France, Paris

    Google Scholar 

  • Zalinescu C (2002) Convex analysis in general vector spaces. World Scientific, Singapore

    Book  Google Scholar 

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Authors and Affiliations

  1. Faculté des Sciences, Département de Mathématiques, Université d’Avignon, 33 rue Louis Pasteur, 84000, Avignon, France

    M. Volle

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  1. M. Volle
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Correspondence to M. Volle.

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In Honor of Professor Marco A. López on his 60th birthday.

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Volle, M. A primal–dual operation on sets linked with closed convex relaxation processes. TOP 20, 534–546 (2012). https://doi.org/10.1007/s11750-011-0211-y

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  • DOI: https://doi.org/10.1007/s11750-011-0211-y

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