IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v313y2024i3p841-857.html
   My bibliography  Save this article

Quadratic regularization of bilevel pricing problems and application to electricity retail markets

Author

Listed:
  • Jacquet, Quentin
  • van Ackooij, Wim
  • Alasseur, Clémence
  • Gaubert, Stéphane

Abstract

We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers’ choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer’s response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer’s response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case.

Suggested Citation

  • Jacquet, Quentin & van Ackooij, Wim & Alasseur, Clémence & Gaubert, Stéphane, 2024. "Quadratic regularization of bilevel pricing problems and application to electricity retail markets," European Journal of Operational Research, Elsevier, vol. 313(3), pages 841-857.
    • Handle: RePEc:eee:ejores:v:313:y:2024:i:3:p:841-857
    DOI: 10.1016/j.ejor.2023.05.006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221723003508
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2023.05.006?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. Fisk, Caroline, 1980. "Some developments in equilibrium traffic assignment," Transportation Research Part B: Methodological, Elsevier, vol. 14(3), pages 243-255, September.
    2. Martine Labbé & Patrice Marcotte & Gilles Savard, 1998. "A Bilevel Model of Taxation and Its Application to Optimal Highway Pricing," Management Science, INFORMS, vol. 44(12-Part-1), pages 1608-1622, December.
    3. repec:cup:cbooks:9780521747387 is not listed on IDEAS
    4. Hongmin Li & Woonghee Tim Huh, 2011. "Pricing Multiple Products with the Multinomial Logit and Nested Logit Models: Concavity and Implications," Manufacturing & Service Operations Management, INFORMS, vol. 13(4), pages 549-563, October.
    5. Elizabeth Baldwin & Paul Klemperer, 2019. "Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities," Econometrica, Econometric Society, vol. 87(3), pages 867-932, May.
    6. Csereklyei, Zsuzsanna, 2020. "Price and income elasticities of residential and industrial electricity demand in the European Union," Energy Policy, Elsevier, vol. 137(C).
    7. Afşar, Sezin & Brotcorne, Luce & Marcotte, Patrice & Savard, Gilles, 2016. "Achieving an optimal trade-off between revenue and energy peak within a smart grid environment," Renewable Energy, Elsevier, vol. 91(C), pages 293-301.
    8. Thomas Kleinert & Martin Schmidt, 2023. "Why there is no need to use a big-M in linear bilevel optimization: a computational study of two ready-to-use approaches," Computational Management Science, Springer, vol. 20(1), pages 1-12, December.
    9. Hongmin Li & Scott Webster & Nicholas Mason & Karl Kempf, 2019. "Product-Line Pricing Under Discrete Mixed Multinomial Logit Demand," Service Science, INFORMS, vol. 21(1), pages 14-28, January.
    10. Youngdae Kim & Sven Leyffer & Todd Munson, 2020. "MPEC Methods for Bilevel Optimization Problems," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 335-360, Springer.
    11. 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.
    12. Sven Leyffer, 2006. "Complementarity constraints as nonlinear equations: Theory and numerical experience," Springer Optimization and Its Applications, in: Stephan Dempe & Vyacheslav Kalashnikov (ed.), Optimization with Multivalued Mappings, pages 169-208, Springer.
    13. Aussel, Didier & Brotcorne, Luce & Lepaul, Sébastien & von Niederhäusern, Léonard, 2020. "A trilevel model for best response in energy demand-side management," European Journal of Operational Research, Elsevier, vol. 281(2), pages 299-315.
    14. Francisco Jara-Moroni & John E. Mitchell & Jong-Shi Pang & Andreas Wächter, 2020. "An enhanced logical benders approach for linear programs with complementarity constraints," Journal of Global Optimization, Springer, vol. 77(4), pages 687-714, August.
    15. Alan Washburn, 2014. "Two-Person Zero-Sum Games," International Series in Operations Research and Management Science, Springer, edition 4, number 978-1-4614-9050-0, March.
    16. S. Dempe & J. F. Bard, 2001. "Bundle Trust-Region Algorithm for Bilinear Bilevel Programming," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 265-288, August.
    17. Train,Kenneth E., 2009. "Discrete Choice Methods with Simulation," Cambridge Books, Cambridge University Press, number 9780521747387, November.
    18. Lijie Bai & John Mitchell & Jong-Shi Pang, 2013. "On convex quadratic programs with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 54(3), pages 517-554, April.
    19. François Gilbert & Patrice Marcotte & Gilles Savard, 2015. "A Numerical Study of the Logit Network Pricing Problem," Transportation Science, INFORMS, vol. 49(3), pages 706-719, August.
    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. Woonghee T. Huh & Hongmin Li, 2023. "Product‐line pricing with dual objective of profit and consumer surplus," Production and Operations Management, Production and Operations Management Society, vol. 32(4), pages 1223-1242, April.
    2. Hongmin Li & Scott Webster & Gwangjae Yu, 2020. "Product Design Under Multinomial Logit Choices: Optimization of Quality and Prices in an Evolving Product Line," Manufacturing & Service Operations Management, INFORMS, vol. 22(5), pages 1011-1025, September.
    3. Xiaobo Li & Hailong Sun & Chung Piaw Teo, 2022. "Convex Optimization for Bundle Size Pricing Problem," Management Science, INFORMS, vol. 68(2), pages 1095-1106, February.
    4. Xi Chen & Zachary Owen & Clark Pixton & David Simchi-Levi, 2022. "A Statistical Learning Approach to Personalization in Revenue Management," Management Science, INFORMS, vol. 68(3), pages 1923-1937, March.
    5. Panos Kouvelis & Yixuan Xiao & Nan Yang, 2015. "PBM Competition in Pharmaceutical Supply Chain: Formulary Design and Drug Pricing," Manufacturing & Service Operations Management, INFORMS, vol. 17(4), pages 511-526, October.
    6. Thomas Loots & Arnoud V. den Boer, 2023. "Data‐driven collusion and competition in a pricing duopoly with multinomial logit demand," Production and Operations Management, Production and Operations Management Society, vol. 32(4), pages 1169-1186, April.
    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. Guang Li & Paat Rusmevichientong & Huseyin Topaloglu, 2015. "The d -Level Nested Logit Model: Assortment and Price Optimization Problems," Operations Research, INFORMS, vol. 63(2), pages 325-342, April.
    9. Aydın Alptekinoğlu & John H. Semple, 2016. "The Exponomial Choice Model: A New Alternative for Assortment and Price Optimization," Operations Research, INFORMS, vol. 64(1), pages 79-93, February.
    10. Ruxian Wang & Maqbool Dada & Ozge Sahin, 2019. "Pricing Ancillary Service Subscriptions," Management Science, INFORMS, vol. 65(10), pages 4712-4732, October.
    11. Zhou, Yong-Wu & Lin, Xiaogang & Zhong, Yuanguang & Xie, Wei, 2019. "Contract selection for a multi-service sharing platform with self-scheduling capacity," Omega, Elsevier, vol. 86(C), pages 198-217.
    12. Dussault, Jean-Pierre & Marcotte, Patrice & Roch, Sebastien & Savard, Gilles, 2006. "A smoothing heuristic for a bilevel pricing problem," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1396-1413, November.
    13. Maxime C. Cohen & Renyu Zhang, 2022. "Competition and coopetition for two‐sided platforms," Production and Operations Management, Production and Operations Management Society, vol. 31(5), pages 1997-2014, May.
    14. Rui Chen & Hai Jiang, 2020. "Capacitated assortment and price optimization under the nested logit model," Journal of Global Optimization, Springer, vol. 77(4), pages 895-918, August.
    15. Teodora Dan & Patrice Marcotte, 2019. "Competitive Facility Location with Selfish Users and Queues," Operations Research, INFORMS, vol. 67(2), pages 479-497, March.
    16. Nakayama, Shoichiro & Chikaraishi, Makoto, 2015. "Unified closed-form expression of logit and weibit and its extension to a transportation network equilibrium assignment," Transportation Research Part B: Methodological, Elsevier, vol. 81(P3), pages 672-685.
    17. Pol Boada-Collado & Victor Martínez-de-Albéniz, 2020. "Estimating and Optimizing the Impact of Inventory on Consumer Choices in a Fashion Retail Setting," Manufacturing & Service Operations Management, INFORMS, vol. 22(3), pages 582-597, May.
    18. Claudia Castaldi & Paolo Delle Site & Francesco Filippi, 2019. "Stochastic user equilibrium in the presence of state dependence," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 8(5), pages 535-559, December.
    19. Hong, Sung-Pil & Kim, Kyung min & Byeon, Geunyeong & Min, Yun-Hong, 2017. "A method to directly derive taste heterogeneity of travellers’ route choice in public transport from observed routes," Transportation Research Part B: Methodological, Elsevier, vol. 95(C), pages 41-52.
    20. Joaquim Dias Garcia & Guilherme Bodin & Alexandre Street, 2024. "BilevelJuMP.jl: Modeling and Solving Bilevel Optimization Problems in Julia," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 327-335, March.

    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:eee:ejores:v:313:y:2024:i:3:p:841-857. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    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.