Abstract

In the infinite dimensional setting, we provide a general formula for the optimal set of a relaxed minimization problem in terms of the approximate minima of the data function. Various applications to the ε-subdifferential calculus are also given.

Author information

Contact details are reproduced from the original publication and may be historical.

Marco A. López

Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain

marco.antonio@ua.es

Keywords

  • Relaxed minimization problem
  • approximate minima
  • epsilon-subdifferential calculus
  • Legendre-Fenchel conjugate

Mathematics Subject Classification

90C48, 90C46; 49N15, 90C25

References

  1. J. Benoist, J.-B. Hiriart-Urruty: What is the subdifferential of the closed convex hull of a function?, SIAM J. Math. Anal. 27 (1996) 1661–1679.https://doi.org/10.1137/S0036141094265936Google Scholar
  2. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).Google Scholar
  3. E. Ernst, M. Thèra, M. Volle: Optimal boundedness criteria for extended-real-valued functions, Optimization 56 (2007) 323–338.https://doi.org/10.1080/02331930600819779Google Scholar
  4. A. Hantoute, M. A. López, C. Zălinescu: Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions, SIAM J. Optim. 19 (2008) 863–882.https://doi.org/10.1137/070700413Google Scholar
  5. J.-B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms. Part I: Fundamentals; Part II: Advanced Theory and Bundle Methods, Springer, Berlin (1993).https://doi.org/10.1007/978-3-662-06409-2Google Scholar
  6. J.-B. Hiriart-Urruty, M. Moussaoui, A. Seeger, M. Volle: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey, Nonlinear Anal. 24 (1995) 1727–1754.https://doi.org/10.1016/0362-546X(94)00221-3Google Scholar
  7. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  8. M. Valadier: Sous-differentiels d’une borne supérieure et d’une somme continue de fonctions convexes, C. R. Acad. Sci., Paris, Sér. A 268 (1969) 39–42.Google Scholar
  9. M. Volle: Sous-différentiel d’une enveloppe supérieure de fonctions convexes, C. R. Acad. Sci., Paris, Sér. I 317 (1993) 845–849.Google Scholar
  10. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

M. A. López, M. Volle. “A Formula for the Set of Optimal Solutions of a Relaxed Minimization Problem. Applications to Subdifferential Calculus.” Journal of Convex Analysis 17 (2010), No. 3&4, 1057–1075. https://doi.org/10.68381/jca17065

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.