Skip to main content
Log in

Concave duality: Application to problems dealing with difference of functions

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper a duality is introduced in a concave sense and its relationship with Toland's duality is studied along with several formulas dealing with the conjugates of differences of convex functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. E. Asplund, “Fréchet differentiability of convex functions,”Acta Mathematica 121 (1968) 31–47.

    Google Scholar 

  2. E. Asplund, “Differentiability of the metric projection in finite dimensional euclidean space,”Proceedings of the American Mathematical Society 38 (1973) 218–219.

    Google Scholar 

  3. J.P. Aubin,L'Analyse Non Linéaire et ses Motivations Economiques (Masson, Paris, 1984).

    Google Scholar 

  4. G. Auchmuty, “Duality for nonconvex variational principles,”Journal of Differential Equations 50 (1983) 80–145.

    Google Scholar 

  5. R. Courant and D. Hilbert,Methods of Mathematical Physics (Interscience, New York, 1953).

    Google Scholar 

  6. J.P. Crouzeix, “Contribution à l'étude des fonctions quasi-convexes,” Thèse Université Clermont-Ferrand II (1977).

  7. I. Ekeland and R. Teman,Analyse Convexe et Problèmes Variationnels (Dunod, Paris, 1974).

    Google Scholar 

  8. I. Ekeland and T. Turnbull,Infinite Dimensional Optimization and Convexity (Chicago Lectures in Mathematics, 1983).

  9. R. Ellaia, “Contribution à l'analyse et l'optimisation de différence de fonctions convexes,” Thèse 3ème cycle Université Paul Sabatier Toulouse (1984).

  10. E. Giner, “Ensembles et fonctions étoilés,” Manuscript 1981.

  11. J.B. Hiriart-Urruty, “Generalized differentiability, duality and optimization for problems dealing with difference of convex functions,” to appear in Lectures Notes in Mathematics.

  12. J.B. Hiriart-Urruty, “A general formula on the conjugate of difference of functions,”Canadian Mathematical Bulletin 29 (1986) 482–485.

    Google Scholar 

  13. J.L. Joly and P.J. Laurent, “Stability and duality in convex minimization problems,”Revue Française d'Informatique et de Recherche opérationnelle 5ème année R-2 (1971) 3–42.

    Google Scholar 

  14. P.J. Laurent,Approximation et Optimisation (Hermann, Paris, 1972).

    Google Scholar 

  15. J.J. Moreau,Fonctionnelles Convexes (Cours au Collège de France, 1966).

  16. J.J. Moreau, “Inf-convolution, sous additivité, convexité des fonctions numériques,”Journal de Mathématiques Pures et Appliquées 49 (1970) 109–154.

    Google Scholar 

  17. J.P. Penot, “Variations on the theme of nonsmooth analysis,” Workshop on Nonlinear Optimization (Sopron, 1984).

  18. L. Pontrjagin, “Linear differential games II,”Doklady Akademii Nauk SSSR 175 (1967) 764–776.

    Google Scholar 

  19. B.N. Pshenichnyi, “Leçons sur les Jeux Différentiels, Contrôle Optimal et Jeux Différentiels,”Cahiers de l'IRIA 4 (1971) 145–226.

    Google Scholar 

  20. R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).

    Google Scholar 

  21. R.T. Rockafellar,Conjugate Duality and Optimization (SIAM, Philadelphia, 1974).

    Google Scholar 

  22. I. Singer, “A Fenchel-Rockafellar type duality theorem for maximization,”Bulletin of the Australian Mathematical Society 20 (1979) 193–198.

    Google Scholar 

  23. I. Singer, “Maximization of lower semi-continuous convex functionals on bounded subsets of locally convex spaces II: Quasi-Lagrangians duality theorem,”Results in Mathematics 3 (1980) 235–248.

    Google Scholar 

  24. Y. Sonntag, “Dérivabilité de la fonction distance d'un point à un convexe,” Multigraphie de l'Université de Provence Marseille (1980).

  25. J.F. Toland, “Duality in nonconvex optimization,”Archive for Rational Mechanics and Analysis 71 (1979) 41–61.

    Google Scholar 

  26. J.F. Toland, “Duality in nonconvex optimization,”Journal of Mathematical Analysis and Applications 56 (1978) 399–415.

    Google Scholar 

  27. J.F. Toland, “On subdifferential caculus and duality in nonconvex optimization,”Bulletin de la Société Mathématique de France 60 (1979) 177–183.

    Google Scholar 

  28. H. Tuy, “A general deterministic approach to global optimization via d-c-programming,” preprint 1985.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Volle, M. Concave duality: Application to problems dealing with difference of functions. Mathematical Programming 41, 261–278 (1988). https://doi.org/10.1007/BF01580767

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01580767

Key words

Navigation