Advertisement

Applied Mathematics and Optimization

, Volume 32, Issue 3, pp 213–234 | Cite as

New results in subdifferential calculus with applications to convex optimization

  • G. Romano
Article

Abstract

Chain and addition rules of subdifferential calculus are revisited in the paper and new proofs, providing local necessary and sufficient conditions for their validity, are presented. A new product rule pertaining to the composition of a convex functional and a Young function is also established and applied to obtain a proof of Kuhn-Tucker conditions in convex optimization under minimal assumptions on the data. Applications to plasticity theory are briefly outlined in the concluding remarks.

Key words

Convex analysis Subdifferentials Chain rules Optimization 

AMS classification

Primary 4902 Secondary 49J52 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Clarke FH (1973) Necessary conditions for nonsmooth problems in optimal control and the calculus of variations. Thesis, University of Washington, Seattle, WAGoogle Scholar
  2. 2.
    Clarke FH (1983) Optimization and Nonsmooth Analysis. Wiley, New YorkGoogle Scholar
  3. 3.
    Ekeland I, Temam R (1974) Analyse convexe et problèmes variationnels. Dunod, ParisGoogle Scholar
  4. 4.
    Ioffe AD, Tihomirov VM (1979) The Theory of Extremal Problems. Nauka, Moscow (English translation, North-Holland, Amsterdam)Google Scholar
  5. 5.
    Laurent PJ (1972) Approximation et Optimisation. Hermann, ParisGoogle Scholar
  6. 6.
    McLinden L (1973) Dual operations on saddle functions. Trans Amer Math Soc 179:363–381Google Scholar
  7. 7.
    Moreau JJ (1966) Fonctionelles convexes. Lecture notes, séminaire: équationes aux dérivées partielles, Collège de FranceGoogle Scholar
  8. 8.
    Moreau JJ (1973) On unilateral constraints, friction and plasticity. New Variational Techniques in Mathematical Physics, CIME Bressanone, Ed. Cremonese, Roma, pp 171–322Google Scholar
  9. 9.
    Panagiotopoulos PD (1985) Inequality Problems in Mechanics and Applications. Birkhäuser, BostonGoogle Scholar
  10. 10.
    Rockafellar RT (1963) Convex functions and dual extremum problems. Thesis Harvard, MAGoogle Scholar
  11. 11.
    Rockafellar RT (1970) Convex Analysis. Princeton University Press, Princeton, NJGoogle Scholar
  12. 12.
    Rockafellar RT (1979) Directionally Lipschitzian functions and subdifferential calculus. Proc London Math Soc (3) 39:331–355Google Scholar
  13. 13.
    Rockafellar RT (1980) Generalized directional derivatives and subgradients of nonconvex functions. Canad J Math XXXII(2):257–280Google Scholar
  14. 14.
    Rockafellar RT (1981) The Theory of Subgradients and its Applications to Problems of Optimization. Convex and Nonconvex Functions. Heldermann Verlag, BerlinGoogle Scholar
  15. 15.
    Romano G, Rosati L, Marotti de Sciarra F (1992) An internal variable theory of inelastic behaviour derived from the uniaxial rigid-perfectly plastic law. Internat J Engrg Sci 31(8):1105–1120Google Scholar
  16. 16.
    Romano G, Rosati L, Marotti de Sciarra F (1993) Variational principles for a class of finite-step elasto-plastic problems with non-linear mixed hardening. Comput Methods Appl Mech Engrg 109:293–314Google Scholar
  17. 17.
    Romano G, Rosati L, Marotti de Sciarra F (1993) A variational theory for finite-step elasto-plastic problems. Internat J Solids 30(17):2317–2334Google Scholar
  18. 18.
    Slater M (1950) Lagrange multipliers revisited: a contribution to non-linear programming. Cowles Commission Discussion Paper, Math 403Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • G. Romano
    • 1
  1. 1.Dipartimento di Scienza delle CostruzioniUniversità di Napoli Federico IINapoliItalia

Personalised recommendations