Skip to main content
SpringerLink
Log in

Single-Leader-Radner-Equilibrium: A New Approach for a Class of Bilevel Problems Under Uncertainty

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

Abstract

Bilevel problems with several followers, often called Single-Leader-Multi-Follower problems, have been proved to be very useful for modeling hierarchical interactions between agents in economics, industry, etc. When uncertainty must be taken into account, a classical approach is to use stochastic bilevel optimization. In this work, we introduce an alternative approach intrinsically integrating at the same time uncertain future and time-dependent decision processes. It is called Single-Leader-Radner-Equilibrium (SLRE) and is characterized by a hierarchical structure with one leader and several followers competing to reach a Radner equilibrium. A variational reformulation of the quasiconcave SLRE model (that is, where the objective function of the followers is only quasiconcave) is proposed and used to prove the existence of an optimistic solution of the quasiconcave SLRE. Finally, thanks to these developments we present a new approach of optimal design of eco-industrial parks.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. For further details, the interested reader can refer to [29] and the references therein.

References

  1. Alexander, B., Barton, G., Petrie, J., Romagnoli, J.: Process Synthesis and Optimisation Tools for Environmental Design: Methodology and Structure. Comput. Chem. Eng. 24, 1195–1200 (2000)

    Article  Google Scholar 

  2. Ansari, Q.H., Kobis, E., Yao, J.: Vector variational inequalities and vector optimization. Springer, Berlin (2018)

    Book  Google Scholar 

  3. Aussel, D.: New developments in quasiconvex optimization. In: Al-Mezel, S.A.R.F., Al-Solamy, R.M., Ansari, Q.H. (eds.) Fixed point theory, variational analysis, and optimization, pp. 171–206. Chapman and Hall/CRC Press, New York (2014)

    Google Scholar 

  4. Aussel, D., Cotrina, J.: Quasimonotone quasivariational inequalities: existence results and applications. J. Optim. Theory Appl. 158(3), 637–652 (2013)

    Article  MathSciNet  Google Scholar 

  5. Aussel, D., Donato, M.B., Milasi, M., Sultana, A.: Existence results for quasi-variational inequalities with applications to radner equilibrium problems. Set-Valued Var. Anal. 29, 931–948 (2021)

    Article  MathSciNet  Google Scholar 

  6. Aussel, D., Van Cao, K., Salas, D.: Optimal design of exchange water networks with control inputs in eco-industrial parks. Energy Econ. 120, 106480 (2023)

    Article  Google Scholar 

  7. Aussel, D., Svensson, A.: A short state of the art on multi-leader-follower games. In: Dempe, S., Zemkoho, A. (eds.) Bilevel Optimization. Springer Optimization and Its Applications, vol. 161, pp. 53–76. Springer, Cham (2021)

    Chapter  Google Scholar 

  8. Aussel, D., Svensson, A.: Is pessimistic bilevel programming a special case of a mathematical program with complementarity constraints? J. Optim. Theory Appl. 181, 504–520 (2019)

    Article  MathSciNet  Google Scholar 

  9. Aussel, D., Svensson, A.: Some remarks about existence of equilibria, and the validity of the EPCC reformulation for multi-leader-follower games. J. Nonlinear Convex Anal. 19(7), 1141–1162 (2019)

    MathSciNet  Google Scholar 

  10. Berge, C.: Topological space. Oliver & Boyd, Edinburgh (1963)

    Google Scholar 

  11. Burtscheidt, J., Claus, M.: Bilevel linear optimization under uncertainty. Advances and Next Challenges, Bilevel Optimization, pp. 485–511 (2020)

  12. Burtscheidt, J., Claus, M., Dempe, S.: Risk-averse models in bilevel stochastic linear programming. SIAM J. Optimiz. 30(1), 377–406 (2020)

    Article  MathSciNet  Google Scholar 

  13. Dempe, S.: Foundations of Bilevel programming. Springer, Berlin (2002)

    Google Scholar 

  14. Dempe, S.: Bilevel optimization: theory, algorithms, applications and a bibliography. Chapter in Bilevel optimization. Springer, Berlin (2020)

    Book  Google Scholar 

  15. Donato, M.B., Milasi, M., Villanacci, A.: Variational formulation of a general equilibrium model with incomplete financial markets and numeraire assets: existence. J. Optim. Theory Appl. 179, 425–451 (2018)

    Article  MathSciNet  Google Scholar 

  16. Heitsch, H., Henrion, R., Kleinert, T., Schmidt, M.: On convex lower-level black-box constraints in bilevel optimization with an application to gas market with chance constraints. J. Glob. Optim. 84(3), 1573–2916 (2022)

    Article  MathSciNet  Google Scholar 

  17. Henrion, R., Surowiec, T.: On calmness conditions in convex bilevel programming. Appl. Anal. 90(6), 951–970 (2011)

    Article  MathSciNet  Google Scholar 

  18. Kreps, D.M.: Microeconomic Foundations I. Princeton University Press, Princeton (2013)

    Google Scholar 

  19. Limosani, M., Milasi, M., Scopelliti, D.: Deregulated electricity market, a stochastic variational approach. Energy Econ. 103, 105493 (2021)

    Article  Google Scholar 

  20. Lucchetti, R.: Convexity and Well-Posed Problems. Springer, Berlin (2006)

    Book  Google Scholar 

  21. Milasi, M., Scopelliti, D.: A stochastic variational approach to study economic equilibrium problems under uncertainty. J. Math. Anal. Appl. 502(1), 125243 (2021)

    Article  MathSciNet  Google Scholar 

  22. Milasi, M., Scopelliti, D.: A variational approach to the maximization of preferences without numerical representation. J. Optim. Theory Appl. 190(3), 879–893 (2021)

    Article  MathSciNet  Google Scholar 

  23. Ramos, M.: Bilevel optimization of eco-industrial parks for the design of sustainable resource networks, Ph.D. thesis (2016)

  24. Ramos, M.A., Boix, M., Aussel, D., Montastruc, L., Vilamajo, P., Domenech, S.: Water exchanges in eco-industrial parks through multiobjective optimization and game theory. Comput. Aided Chem. Eng. 37, 1997–2002 (2015)

    Article  Google Scholar 

  25. Ramos, M., Rocafull, M., Boix, M., Aussel, D., Montastruc, L., Domenech, S.: Utility network optimization in eco-industrial parks by a multi-leader-follower game methodology. Comput. Chem. Eng. 112, 132–153 (2018)

    Article  Google Scholar 

  26. Salas, D., Svensson, A.: Existence of solution for deterministic bilevel games under a general Bayesian approach. SIAM J. 33(3), 2311–2340 (2023)

    Article  MathSciNet  Google Scholar 

  27. Salas, D., Van Cao, K., Aussel, D., Montastruc, L.: Optimal design of exchange networks with blind inputs and its application to eco-industrial parks. Comput. Chem. Eng. 143, 1–18 (2020)

    Article  Google Scholar 

  28. Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation. Optim. 57(3), 395–418 (2008)

    Article  Google Scholar 

  29. Simon, C.P., Blume, L.E.: Mathematics for Economists. W.W Norton & Company Inc, New York (1994)

    Google Scholar 

  30. Tan, N.X.: Quasi-variational inequality in topological linear locally convex Hausdorff spaces. Math. Nachr. 122, 231–245 (1985)

    Article  MathSciNet  Google Scholar 

  31. Von Stackelberg, H.: Marktform und Gleichgewicht. Springer, Berlin (1934)

    Google Scholar 

Download references

Acknowledgements

This research benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research” and from the support to this program from EDF.

Author information

Authors and Affiliations

  1. Department of Economics and Management, University of Brescia, Brescia, Italy

    Elisabetta Allevi, Rossana Riccardi & Domenico Scopelliti

  2. Lab. PROMES UPR CNRS 8521, Université de Perpignan Via Domitia, Perpignan, France

    Didier Aussel

Authors
  1. Elisabetta Allevi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Didier Aussel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rossana Riccardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Domenico Scopelliti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Didier Aussel.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Martin Schmidt.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

D. Scopelliti would like to thank the Laboratoire PROMES of the University of Perpignan and Prof. Didier Aussel for their hospitality. Indeed, this work has been prepared, while this author was visiting researcher in this laboratory.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allevi, E., Aussel, D., Riccardi, R. et al. Single-Leader-Radner-Equilibrium: A New Approach for a Class of Bilevel Problems Under Uncertainty. J Optim Theory Appl 200, 344–370 (2024). https://doi.org/10.1007/s10957-023-02339-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02339-5

Keywords

Search

Navigation