Skip to main content
SpringerLink
Log in

The boundedness of the Lagrange multipliers set and duality in mathematical programming

  • Published:
Zeitschrift für Operations Research Aims and scope Submit manuscript

Abstract

We are interested in this paper to determine the properties which are, in the primal, related to the boundedness properties of the set of the Lagrange multipliers. In convex programming it is shown that it is more or less equivalent to the generealized Slater condition. From there, we generalize to Banach spaces all the results on this topic which were known for finite dimensional spaces in differentiable and locally Lipschitz programming.

Zusammenfassung

Wir untersuchen in dieser Arbeit Eigenschaften des primalen Problems im Zusammenhang mit der Beschränktheit der Menge der Lagrange-Multiplikatoren. Bei konvexen Programmen ist dies im wesentlichen gleichwertig mit der verallgemeinerten Slater-Bedingung. Von daher verallgemeinern wir alle einschlägigen Resultate aus der differenzierbaren oder der lokal Lipschitz-stetigen Optimierung von endlichdimensionalen Räumen auf Banach-Räume.

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

  • Auslender, A.: Stabilité différentiable en programmation non convexe non différentiable, C.R. Acad. Sci. Paris286, 1978, 575–577.

    Google Scholar 

  • Borwein, J.M.: Convex relations in analysis and optimisation, Carnegie-Mellon University D.P. 5, 1980.

  • Brokate, M.: A regularity condition for optimization in Banach spaces: counter examples. Appl. Math. Optim.6, 1980, 189–192.

    Google Scholar 

  • Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Research1, 1976, 165–174.

    Google Scholar 

  • Gauvin, J.: A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Progr.12, 1977, 136–138.

    Google Scholar 

  • Gauvin, J., andJ. W. Tolle: Differential stability in nonlinear programming. SIAM J. on Control15, 1977, 294–311.

    Google Scholar 

  • Hiriart-Urruty, J.-B.: Contributions à la programmation mathématique: cas deterministe et stochastique. Clermont-Ferrand II University Thesis, 1977.

  • —: On optimality conditions in nondifferentiable programming. Math. Progr.14, 1978, 73–86.

    Google Scholar 

  • Lévine, P., andJ.-Ch. Pomerol: Sufficient conditions for Kuhn-Tucker vectors in convex programmin. SIAM J. on Control17, 1979, 689–699.

    Google Scholar 

  • —: Erratum. SIAM J. on Control19, 1981, 431–432.

    Google Scholar 

  • Michel, Ph.: Problème des inégalités. Application à la programmation et au contrôle optimal. Bull. Soc. Math. France101, 1973, 413–439.

    Google Scholar 

  • —: Problème des inégalités et application à la programmation dans le cas où l'espace d'arrivée est de dimension finie. C.R. Acad. Sci. Paris278, 1974, 389–391.

    Google Scholar 

  • Nguyen, V.H., J.J. Strodiot andR. Mifflin: On conditions to have bounded multipliers in locally Lipschitz programming. Math. Progr.18, 1980, 100–106.

    Google Scholar 

  • Penot, J.-P.: On the existence of Lagrange multipliers in nonlinear programming in Banach spaces. Proc. Symp. on Optimization and Control theory, Oberwolfach 1980a.

  • -: On regularity conditions in mathematical programming. Pau University D.P., 1980b. To appear in Math. Progr. Studies.

  • Pomerol, J.-Ch.: Sufficient conditions for primal dual values equality in convex programming. Operations Research Verfahren31, 1979a, 517–524.

    Google Scholar 

  • —: Application de la programmation convexe à la programmation différentiable. C.R. Acad. Sci. Paris288, 1979b, 1041–1044.

    Google Scholar 

  • —: Application de la programmation convexe à la programmation non différentiable, C.R. Acad. Sci. Paris289, 1979c, 805–808.

    Google Scholar 

  • —: Nondifferentiable programming via convex programming. Methods of Operations Research37, 1980a, 213–220.

    Google Scholar 

  • -: Contribution à la programmation mathématique: existence de multiplicateurs de Lagrange et stabilité. P. & M. Curie University Thesis, Paris 1980b.

  • Robinson, S.M.: Stability theory for systems of inqualities, part II:differentiable nonlinear systems. SIAM J. Numer. Anal.13, 1976, 497–513.

    Google Scholar 

  • Rockafellar, R.T.: Convex Analysis. Princeton 1970.

  • -: Conjugate duality and optimization. CBS/NSF regional conference series in applied mathematics 16. Ed. by SIAM, Philadelphia 1974.

  • Thibault, L.: Mathematical programming and optimal control problems defined by compactly Lipschitzian mappings. Séminaire d'analyse convexe, exposé n∘ 10. Montpellier 1978.

  • Zowe, J., andS. Kurcyusz: Regularity and stability for mathematical programming problem in Banach spaces. Appl. Math. Optim.5, 1979, 49–62.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire d'Econométrie de l'Université P. et M. Curie, 4 Place Jussieu, F-75230, Paris cedex 5

    J. -Ch. Pomerol

Authors
  1. J. -Ch. Pomerol
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pomerol, J.C. The boundedness of the Lagrange multipliers set and duality in mathematical programming. Zeitschrift für Operations Research 25, 191–204 (1981). https://doi.org/10.1007/BF01917172

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01917172

Keywords

Search

Navigation