IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v206y2025i1d10.1007_s10957-025-02683-8.html
   My bibliography  Save this article

A Regularized Variance-Reduced Modified Extragradient Method for Stochastic Hierarchical Games

Author

Listed:
  • Shisheng Cui

    (Beijing Institute of Technology)

  • Uday V. Shanbhag

    (University of Michigan)

  • Mathias Staudigl

    (University of Mannheim)

Abstract

We consider an N-player hierarchical game in which the ith player’s objective comprises of an expectation-valued term, parametrized by rival decisions, and a hierarchical term. Such a framework allows for capturing a broad range of stochastic hierarchical optimization problems, Stackelberg equilibrium problems, and leader-follower games. We develop an iteratively regularized and smoothed variance-reduced modified extragradient framework for iteratively approaching hierarchical equilibria in a stochastic setting. We equip our analysis with rate statements, complexity guarantees, and almost-sure convergence results. We then extend these statements to settings where the lower-level problem is solved inexactly and provide the corresponding rate and complexity statements. Our model framework encompasses many game theoretic equilibrium problems studied in the context of power markets. We present a realistic application to the study of virtual power plants, emphasizing the role of hierarchical decision making and regularization. Preliminary numerics suggest that empirical behavior compares well with theoretical guarantees.

Suggested Citation

  • Shisheng Cui & Uday V. Shanbhag & Mathias Staudigl, 2025. "A Regularized Variance-Reduced Modified Extragradient Method for Stochastic Hierarchical Games," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-53, July.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:1:d:10.1007_s10957-025-02683-8
    DOI: 10.1007/s10957-025-02683-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-025-02683-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-025-02683-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Shisheng Cui & Uday Shanbhag & Mathias Staudigl & Phan Vuong, 2022. "Stochastic relaxed inertial forward-backward-forward splitting for monotone inclusions in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 83(2), pages 465-524, November.
    2. X. M. Hu & D. Ralph, 2004. "Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 365-390, November.
    3. Victor DeMiguel & Huifu Xu, 2009. "A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application," Operations Research, INFORMS, vol. 57(5), pages 1220-1235, October.
    4. Ahmet Alacaoglu & Yura Malitsky & Volkan Cevher, 2021. "Forward-reflected-backward method with variance reduction," Computational Optimization and Applications, Springer, vol. 80(2), pages 321-346, November.
    5. Hanif D. Sherali, 1984. "A Multiple Leader Stackelberg Model and Analysis," Operations Research, INFORMS, vol. 32(2), pages 390-404, April.
    6. Rui, Zhaobiao & Peng, Weicai & Qin, Ximei & Wang, Jun, 2023. "Assessing carbon cap-and-trade policies on hybrid renewable energy investments: Implications for pricing and capacity decisions," Resources Policy, Elsevier, vol. 86(PA).
    7. Beck, Yasmine & Ljubić, Ivana & Schmidt, Martin, 2023. "A survey on bilevel optimization under uncertainty," European Journal of Operational Research, Elsevier, vol. 311(2), pages 401-426.
    8. Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, January.
    9. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    10. Jin, Xin & Baker, Kyri & Christensen, Dane & Isley, Steven, 2017. "Foresee: A user-centric home energy management system for energy efficiency and demand response," Applied Energy, Elsevier, vol. 205(C), pages 1583-1595.
    11. repec:inm:orstsy:v:11:y:2021:i:2:p:112-139 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ludovic A. Julien, 2024. "Noncooperative oligopoly equilibrium in markets with hierarchical competition," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(2), pages 325-371, June.
    2. Julien, Ludovic A., 2017. "On noncooperative oligopoly equilibrium in the multiple leader–follower game," European Journal of Operational Research, Elsevier, vol. 256(2), pages 650-662.
    3. Ming Hu & Masao Fukushima, 2011. "Variational Inequality Formulation of a Class of Multi-Leader-Follower Games," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 455-473, December.
    4. David Pozo & Enzo Sauma & Javier Contreras, 2017. "Basic theoretical foundations and insights on bilevel models and their applications to power systems," Annals of Operations Research, Springer, vol. 254(1), pages 303-334, July.
    5. Desmond Cai & Anish Agarwal & Adam Wierman, 2020. "On the Inefficiency of Forward Markets in Leader–Follower Competition," Operations Research, INFORMS, vol. 68(1), pages 35-52, January.
    6. Victor DeMiguel & Huifu Xu, 2009. "A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application," Operations Research, INFORMS, vol. 57(5), pages 1220-1235, October.
    7. Attila Tasnádi, 2010. "Quantity-setting games with a dominant firm," Journal of Economics, Springer, vol. 99(3), pages 251-266, April.
    8. Lina Mallozzi & Roberta Messalli, 2017. "Multi-Leader Multi-Follower Model with Aggregative Uncertainty," Games, MDPI, vol. 8(3), pages 1-14, June.
    9. Ludovic Julien, 2011. "A note on Stackelberg competition," Journal of Economics, Springer, vol. 103(2), pages 171-187, June.
    10. James Ang & Masao Fukushima & Fanwen Meng & Takahiro Noda & Jie Sun, 2013. "Establishing Nash equilibrium of the manufacturer–supplier game in supply chain management," Journal of Global Optimization, Springer, vol. 56(4), pages 1297-1312, August.
    11. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2021. "A Local Variation Method for Bilevel Nash Equilibrium Problems," CSEF Working Papers 620, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
    12. Ludovic A. Julien & Olivier Musy & Aurélien W. Saidi, 2011. "The Stackelberg equilibrium as a consistent conjectural equilibrium," Economics Bulletin, AccessEcon, vol. 31(1), pages 938-949.
    13. Huifu Xu & Dali Zhang, 2013. "Stochastic Nash equilibrium problems: sample average approximation and applications," Computational Optimization and Applications, Springer, vol. 55(3), pages 597-645, July.
    14. Uday V. Shanbhag & Gerd Infanger & Peter W. Glynn, 2011. "A Complementarity Framework for Forward Contracting Under Uncertainty," Operations Research, INFORMS, vol. 59(4), pages 810-834, August.
    15. Addis Belete Zewde & Semu Mitiku Kassa, 2021. "Multilevel multi-leader multiple-follower games with nonseparable objectives and shared constraints," Computational Management Science, Springer, vol. 18(4), pages 455-475, October.
    16. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2013. "Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 33-64, July.
    17. Daniel Ralph & Huifu Xu, 2011. "Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 568-592, August.
    18. Jian Yao & Ilan Adler & Shmuel S. Oren, 2008. "Modeling and Computing Two-Settlement Oligopolistic Equilibrium in a Congested Electricity Network," Operations Research, INFORMS, vol. 56(1), pages 34-47, February.
    19. Dali Zhang & Huifu Xu & Yue Wu, 2010. "A two stage stochastic equilibrium model for electricity markets with two way contracts," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 1-45, February.
    20. Wu, Ruijia & Van Gorder, Robert A., 2018. "Nonlinear dynamics of discrete time multi-level leader–follower games," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 240-250.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:206:y:2025:i:1:d:10.1007_s10957-025-02683-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.