Skip to main content
SpringerLink
Log in

The lagrange multiplier set and the generalized gradient set of the marginal function of a differentiable program in a Banach space

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We prove that, under the usual constraint qualification and a stability assumption, the generalized gradient set of the marginal function of a differentiable program in a Banach space contains the Lagrange multiplier set. From there, we deduce a sufficient condition in order that, in finite-dimensional spaces, the Lagrange multiplier set be equal to the generalized gradient set of the marginal function.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alt, W.,Lipschitzian Perturbations of Infinite Optimization Problems, Bayreuth University, Report No. 30, 1980.

  2. Auslender, A.,Differentiable Stability in Nonconvex and Nondifferentiable Programming, Mathematical Programming Studies, Vol. 10, pp. 29–41, 1979.

    Google Scholar 

  3. Fiacco, A. V., andHutzler, W. F.,Basic Results in the Development of Sensitivity and Stability Analysis in Nonlinear Programming, Computers and Operations Research, Vol. 9, pp. 9–28, 1982.

    Google Scholar 

  4. Fiacco, A. V.,Optimal Value Differential Stability Bounds under the Mangasarian-Fromovitz Qualification (to appear).

  5. Gauvin, J.,The Generalized Gradient of a Marginal Function in Mathematical Programming, Mathematics of Operations Research, Vol. 4, pp. 458–463, 1979.

    Google Scholar 

  6. Gauvin, J., andTolle, J. W.,Differential Stability in Nonlinear Programming, SIAM Journal on Control and Optimization, Vol. 15, pp. 294–311, 1977.

    Google Scholar 

  7. Gauvin, J., andDubeau, F. Differential Properties of the Marginal Function in Mathematical Programming, Mathematical Programming Studies (to appear).

  8. Janin, R.,First-Order Differential Stability in Nonconvex Programming, Centre Universitaire des Antilles, Pointe à Pitre, Report No. 8101, 1981.

  9. Lempio, F., andMaurer, H.,Differentiable Stability in Infinite-Dimensional Nonlinear Programming, Applied Mathematics and Optimization, Vol. 6, pp. 139–152, 1980.

    Google Scholar 

  10. Penot, J. P.,Differentiability of Relations and Differential Stability of Perturbed Optimization Problems, Pau University, Report No. 7, 1981.

  11. Rockafellar, R. T.,Lagrange Multipliers and Subderivatives of Optimal Value Functions in Nonlinear Programming, Mathematical Programming Studies, Vol. 17, pp. 28–66, 1982.

    Google Scholar 

  12. Robinson, S. M.,Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.

    Google Scholar 

  13. Robinson, S. M.,Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems, SIAM Journal on Numerical Analysis, Vol. 13, pp. 497–513, 1976.

    Google Scholar 

  14. Rockafeller, R. T.,Generalized Directional Derivatives and Subgradient of Nonconvex Functions, Canadian Journal of Mathematics, Vol. 32, pp. 257–280, 1980.

    Google Scholar 

  15. Rockafellar, R. T.,Directionally Lipschitzian Functions and Subdifferential Calculus, Proceedings of the London Mathematical Society, Vol. 39, pp. 331–355, 1979.

    Google Scholar 

  16. Penot, J. P.,On Regularity Conditions in Mathematical Programming, Mathematical Programming Studies (to appear).

  17. Sach, P. H.,Un Théorème Général de Type d'Application Localement Ouverte, Bordeaux I University, Report No. 8021, 1980.

  18. Clarke, F. H.,A New Approach to Lagrange Multipliers, Mathematics of Operations Research, Vol. 1, pp. 165–174, 1976.

    Google Scholar 

  19. Pomerol, J. C.,Application de la Programmation Convexe à la Programmation non Differentiable, Comptes Rendus de l'Académie des Sciences de Paris, Série A, Vol. 289, pp. 805–808, 1979.

    Google Scholar 

  20. Ekeland, I., andTemam, R.,Analyse Convexe et Problèmes Variationnels, Dunod, Paris, France, 1974.

    Google Scholar 

  21. Gauvin, J., andDubeau, F.,Some Examples and Counterexamples for the Stability Analysis of Nonlinear Programming Problems, Ecole Polytechnique de Montréal, Report No. 5, 1981.

Download references

Author information

Authors and Affiliations

  1. Laboratory of Econometrics, University P. and M. Curie, Paris, France

    J. C. Pomerol (Maître Assistant)

Authors
  1. J. C. Pomerol
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. V. Fiacco

The author wishes to thank J. B. Hiriart-Urruty for many helpful suggestions during the preparation of this paper. He also wishes to express his appreciation to the referees for their many valuable comments.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pomerol, J.C. The lagrange multiplier set and the generalized gradient set of the marginal function of a differentiable program in a Banach space. J Optim Theory Appl 38, 307–317 (1982). https://doi.org/10.1007/BF00935341

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00935341

Key Words

Search

Navigation