Skip to main content
SpringerLink
Log in

Necessary optimality conditions for Stackelberg problems

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

Abstract

First-order necessary optimality conditions are derived for a class of two-level Stackelberg problems in which the followers' lower-level problems are convex programs with unique solutions. To this purpose, generalized Jacobians of the marginal maps corresponding to followers' problems are estimated. As illustrative examples, two discretized optimum design problems with elliptic variational inequalities are investigated. The theoretical results may be used also for the numerical solution of the Stackelberg problems considered by nondifferentiable optimization methods.

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. Von Stackelberg, H.,The Theory of the Market Economy, Oxford University Press, Oxford, England, 1952.

    Google Scholar 

  2. Simaan, M., andCruz, J. B.,On the Stackelberg Strategy in Nonzero-Sum Games, Journal of Optimization Theory and Applications, Vol. 11, pp. 535–555, 1973.

    Google Scholar 

  3. Loridan, P., andMorgan, J.,A Theoretical Approximation Scheme for Stackelberg Problems, Journal of Optimization Theory and Applications, Vol. 61, pp. 95–110, 1989.

    Google Scholar 

  4. Basar, T., andOlsder, G. J.,Dynamic Noncooperative Game Theory, Academic Press, New York, New York, 1982.

    Google Scholar 

  5. Shimizu, K., andAiyoshi, E.,A New Computational Method for Stackelberg and Minimax Problems by Use of a Penalty Method, IEEE Transactions on Automatic Control, Vol. 26, pp. 460–466, 1981.

    Google Scholar 

  6. Outrata, J. V.,On the Numerical Solution of a Class of Stackelberg Problems, Zeitschrift für Operations Research, Vol. 4, pp. 255–278, 1990.

    Google Scholar 

  7. Mignot, F.,Contrôle dans les Inéquations Variationelles Elliptiques, Journal of Functional Analysis, Vol. 22, pp. 130–185, 1976.

    Google Scholar 

  8. Mignot, F., andPuel, J. P.,Optimal Control in Some Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 22, pp. 466–476, 1984.

    Google Scholar 

  9. Shi, S. H.,Optimal Control of Strongly Monotone Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 26, pp. 274–290, 1988.

    Google Scholar 

  10. Shi, S. H.,Erratum: Optimal Control of Strongly Monotone Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 28, pp. 243–249, 1990.

    Google Scholar 

  11. Hiriart-Urruty, J. B., andThibault, L.,Existence et Caractérization de Differentielles Généralisées, Comptes Rendus de l'Académie des Sciences de Paris, Vol. 290, pp. 1091–1094, 1980.

    Google Scholar 

  12. Clarke, F. H.,Optimization and Nonsmooth Analysis, J. Wiley and Sons, New York, New York, 1983.

    Google Scholar 

  13. Cornet, B., andLaroque, G.,Lipschitz Properties of Solutions in Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 53, pp. 407–427, 1987.

    Google Scholar 

  14. Jitorntrum, K.,Solution Point Differentiability without Strict Complementarity in Nonlinear Programming, Mathematical Programming Study, Vol. 21, pp. 127–138, 1984.

    Google Scholar 

  15. Kyparisis, J.,Sensitivity Analysis for Nonlinear Programs and Variational Inequalities with Nonunique Multipliers, Mathematics of Operations Research, Vol. 15, pp. 286–298, 1990.

    Google Scholar 

  16. Fiacco, A. V.,Sensitivity Analysis for Nonlinear Programming Using Penalty Methods, Mathematical Programming, Vol. 10, pp. 287–311, 1976.

    Google Scholar 

  17. Malanowski, K.,Differentiability with Respect to Parameters of Solutions to Convex Programming Problems, Mathematical Programming, Vol. 33, pp. 352–361, 1985.

    Google Scholar 

  18. Hiriart-Urruty, J. B.,Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces, Mathematics of Operations Research, Vol. 4, pp. 79–97, 1979.

    Google Scholar 

  19. Kočvara, M., andOutrata, J. V.,A Nondifferentiable Approach to the Solution of Optimum Design Problems with Variational Inequalities, System Modelling and Optimization, Edited by P. Kall, Springer-Verlag, Berlin, Germany, pp. 364–373, 1992.

    Google Scholar 

  20. Haslinger, J., andNeittaanmäki, P.,Finite-Element Approximations for Optimal Shape Design, J. Wiley and Sons, Chichester, England, 1988.

    Google Scholar 

  21. Hlaváček, I.,Shape Optimization of an Elastic Perfectly Plastic Body, Aplikace Matematiky, Vol. 32, pp. 381–400, 1987.

    Google Scholar 

  22. Schramm, H., andZowe, J.,A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results, SIAM Journal on Optimization, Vol. 2, pp. 121–152, 1992.

    Google Scholar 

  23. Schittkowski, K.,nlpql: A fortran Subroutine Solving Constrained Nonlinear Programming Problems, Annals of Operations Research, Vol. 5, pp. 485–500, 1985.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Information Theory and Automation, Prague, Czechoslovakia

    J. V. Outrata (Staff Member)

Authors
  1. J. V. Outrata
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. Simaan

The author would like to thank the referees for their careful review of the manuscript and for the helpful comments and suggestions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Outrata, J.V. Necessary optimality conditions for Stackelberg problems. J Optim Theory Appl 76, 305–320 (1993). https://doi.org/10.1007/BF00939610

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00939610

Key Words

Search

Navigation