Abstract

The asymptotical limit of the trajectory defined by the continuous steepest descent method for a proper closed convex function f on a Hilbert space is characterized in the set of minimizers of f via an asymptotical variational principle of Brezis-Ekeland type. The implicit discrete analogue (prox method) is also considered.

Author information

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

B. Lemaire

Université Montpellier II, Place E. Bataillon, 34095 Montpellier Cedex 5, France

Keywords

  • Asymptotical
  • convex minimization
  • differential inclusion
  • prox method
  • steepest descent
  • variational principle

References

  1. H. Attouch: Viscosity methods. Application to Mechanics and Mathematical Programming, Working paper Université Montpellier II Mathématiques, 1993.Google Scholar
  2. H. Brezis: Opérateurs maximaux monotones, Lecture Note no. 5, North-Holland, 1973.Google Scholar
  3. H. Brezis, I. Ekeland: Un principe variationnel associé à certaines équations paraboliques. Le cas indépendant du temps, C.R. Acad. Sc. Paris, t. 282 (3 mai 1976) Série A, 971–974.Google Scholar
  4. H. Brezis, P.L. Lions: Produits infinis de résolvantes, Israël Journal of Mathematics, Vol 29, no. 4, 1978.https://doi.org/10.1007/BF02761171Google Scholar
  5. R.E Bruck: Asymptotic Convergence of Nonlinear Contration Semigroups in Hilbert Space, Journal of Functional Analysis 18 (1975) 15–26.https://doi.org/10.1016/0022-1236(75)90027-0Google Scholar
  6. O. Güler: On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control and Optimization, Vol 29, no. 2 (1991) 403–419.https://doi.org/10.1137/0329022Google Scholar
  7. B. Lemaire: Régularisation et pénalisation en optimisation convexe, Séminaire d’Analyse Convexe, Institut de Mathématiques USTL, exposé no. 21, 1971.Google Scholar
  8. B. Lemaire: The proximal algorithm, in New Methods in Optimization and their Industrial Uses, Proc. Symp. Pau and Paris 1987, Int. Ser. Numer. Math., Vol 87, Birkhäuser Verlag (1989) 73–89.Google Scholar
  9. B. Lemaire: About the convergence of the proximal method, in Advances in Optimization, Proceedings Lambrecht 1991, Lecture Notes in Economics and Mathematical Systems 382, Springer-Verlag (1992) 39–51.https://doi.org/10.1007/978-3-642-51682-5_4Google Scholar
  10. B. Lemaire: Bounded Diagonally Stationary Sequences in Convex Optimization, Journal of Convex Analysis, Heldermann Verlag Berlin, 1 (1994) 75–86.https://doi.org/10.68381/jca01005Google Scholar
  11. J.J Moreau: Sur les lois de frottement, de plasticité et de viscosité, C.R. Acad. Sci. Paris, Sér. A, 271 (1970) 608–611.Google Scholar
  12. J.J Moreau: Sur l’évolution d’un système élasto-visco-plastique, C.R. Acad. Sci. Paris, Sér. A, 273 (1971) 118–121.Google Scholar
  13. J.J Moreau: Application of Convex Analysis to the treatment of elastoplastic systems, in Proceedings Symposium IUTAM/IMU, edited by P.Germain & B.Nayroles, Lecture Notes in Mathematics, vol. 503, Springer-Verlag (1976) 56–89.https://doi.org/10.1007/BFb0088746Google Scholar
  14. R.T. Rockafellar: Monotone Operators and the proximal point algorithm, SIAM J. Control and Optimisation (1976) 877–898.https://doi.org/10.1137/0314056Google Scholar
  15. A. Tikhonov, V. Arsenine: Méthodes de résolution de problèmes mal posés, M.I.R, 1974.Google Scholar

Suggested citation

B. Lemaire. “An Asymptotical Variational Principle Associated with the Steepest Descent Method for a Convex Function.” Journal of Convex Analysis 3 (1996), No. 1, 63–70. https://doi.org/10.68381/jca03005

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1996. Rights now held by Banach Press.