Skip to main content

Approximate solutions for two-level optimization problems

  • Chapter
Trends in Mathematical Optimization

Abstract

This paper is devoted to general results for approximating two-level optimization problems in which the set of solutions to the lower level problem is not a singleton.

In particular, we give sufficient conditions for upper semicontinuity of ε-Stackelberg solutions by using the notions of convergence presented at “Journées Fermat: Mathematics for Optimization” (Toulouse). ([8]).

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Attouch, H.; Wets, R.: Approximation and convergence in nonlinear optimization. Nonlinear Programming 4, Mangasarian O., Meyer, R., Robinson, S. ( eds. ), Academic Press 1981, 367–394.

    Google Scholar 

  2. Basar, T.; Olsder, G. J.: Dynamic noncooperative game theory. Academic Press, New York, 1982.

    Google Scholar 

  3. de Giorgi, E.; Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei, Vol. 58, No. 8 (1975), 842–850.

    Google Scholar 

  4. de Giorgi, E.; Franzoni, T.: Su un tipo di convergenza variazionale. Rendiconti del Seminario Matematico di Brescia, Vol. 3 (1979), 63–101.

    Google Scholar 

  5. del Prete, I.; Lignola, M. B.: On the variational properties of r (d)-convergence. Pubblicazioni dell’Istituto di Matematica, Napoli, 1981.

    Google Scholar 

  6. Dolecki, S.: Tangency and Differentiation: some applications of convergence theory. Ann. Mat. Pura Appi., Vol. 130 (1982), 223–255.

    Article  Google Scholar 

  7. Loridan, P.; Morgan, J.: Approximation results for a two-level optimization problem and application to penalty methods. Submitted for Publication to Mathematical Programming 1985.

    Google Scholar 

  8. Loridan, P.; Morgan, J.: A general approach for the Stackelberg problem and applications. Journées Fermati Mathematics for Optimization, Toulouse, 1985.

    Google Scholar 

  9. Papavassilopoulos, G.; Cruz, J.: Nonclassical control problems and Stackelberg games. IEEE Transactions on Automatic Control, Vol. AC-24, No. 2 (1979), 155–165.

    Article  Google Scholar 

  10. Robinson, S.; Day, R.: A sufficient condition for continuity of optimal sets in mathematical programming. Journal of Mathemtical Analysis and Applications 45 (1974), 506–511.

    Article  Google Scholar 

  11. Shimizu, K.; Aiyoshi, E.: A new computational method for Stackelberg and min- max problems by use of a penalty method. IEEE Transactions on Automatic Control, Vol. AC-26, No. 2 (1981), 460–466.

    Article  Google Scholar 

  12. Simaan, M.; Cruz, J.: On the Stackelberg strategy in nonzero-sum games. Journal of Optimization Theory and Applications 11, No. 5 (1973), 533–555.

    Article  Google Scholar 

  13. Simaan, M.; Cruz, J.: Additional aspects of the Stackelberg strategy in nonzero- sum games. Journal of Optimization Theory and Applications 11, No. 6 (1973), 613–626.

    Article  Google Scholar 

  14. von Stackelberg, H.: The theory of market economy. Oxford University Press, Oxford 1952.

    Google Scholar 

  15. Zolezzi, T.: On convergence of minima. Bollettino U.M.I., Vol. 8 (1973), 246–257.

    Google Scholar 

  16. Zolezzi, T.: On stability analysis in mathematical programming. Mathematical Programming Study 21 (1984), 227–242.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Birkhäuser Verlag Basel

About this chapter

Cite this chapter

Loridan, P., Morgan, J. (1988). Approximate solutions for two-level optimization problems. In: Hoffmann, KH., Zowe, J., Hiriart-Urruty, JB., Lemarechal, C. (eds) Trends in Mathematical Optimization. International Series of Numerical Mathematics/Internationale Schriftenreihe zur Numerischen Mathematik/Série internationale d’Analyse numérique, vol 84. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9297-1_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9297-1_13

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9984-0

  • Online ISBN: 978-3-0348-9297-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics