Skip to main content
Log in

Contingent derivatives of set-valued maps and applications to vector optimization

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we investigate contingent derivatives of set-valued maps and their lower and upper semidifferentiability properties. We provide also some calculus rules for these derivatives in infinite dimensional spaces. The concept of contingent derivatives is then applied to produce several necessary and sufficient conditions for vector optimization problems with set-valued objectives.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.-P. Aubin, “Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions,” in: L. Nachbin, ed.,Advances in Mathematics Supplementary Studies 7A (Academic Press, New York, 1981) pp. 159–229.

    Google Scholar 

  2. J.-P. Aubin and I. Ekeland,Applied Nonlinear Analysis (Wiley, New York, 1984).

    Google Scholar 

  3. A. Ben-Tal and J. Zowe, “Directional derivatives in nonsmooth optimization,”Journal of Optimization Theory and Applications 47 (1985) 483–490.

    Google Scholar 

  4. G. Bouligand, “Sur l'existence des demi-tangentes à une curbe de Jordan,”Fundamenta Mathematicae 15 (1930) 215–218.

    Google Scholar 

  5. F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).

    Google Scholar 

  6. H.W. Corley, “Existence and Lagrangean duality for maximizations of set-valued functions,”Journal of Optimization Theory and Applications 54 (1987) 489–501.

    Google Scholar 

  7. H. Frankowska, “Adjoint differentiable inclusions in necessary conditions for the minimal trajectories of differential inclusions,”Annales de l'Institut de Henri Poincaré, Analyse non lineaire 2 (1985) 75–99.

    Google Scholar 

  8. D.T. Luc, “Theorems of the alternative and applications in multiobjective optimization,”Acta Mathematica Hungarica 45 (1985) 311–320.

    Google Scholar 

  9. D.T. Luc, “On duality theory in multiobjective programming,”Journal of Optimization Theory and Applications 43 (1984) 557–582.

    Google Scholar 

  10. J.-P. Penot, “Differentiability of relations and differential stability of perturbed optimization problems,”SIAM Journal on Control and Optimization 22 (1984) 529–551.

    Google Scholar 

  11. C. Ursescu, “Tangent sets' calculus and necessary conditions for extremality,”SIAM Journal on Control and Optimization 20 (1982) 563–574.

    Google Scholar 

  12. D.E. Ward and J.M. Borwein, “Nonsmooth calculus in finite dimensions,”SIAM Journal on Control and Optimization 25 (1987) 1312–1340.

    Google Scholar 

  13. C. Zalinescu, “Solvability results for sublinear functions and operators,”Zeitschrift für Operations Research 3 (1987) A79-A101.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This paper was written when the author was at the University of Erlangen-Nurnberg under a grant of the Alexander von Humboldt Foundation.

On leave from the Institute of Mathematics, Hanoi, Vietnam.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luc, D.T. Contingent derivatives of set-valued maps and applications to vector optimization. Mathematical Programming 50, 99–111 (1991). https://doi.org/10.1007/BF01594928

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01594928

AMS Subject Classifications

Key words

Navigation