IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v65y2016i3d10.1007_s10589-016-9851-z.html
   My bibliography  Save this article

Inexact stabilized Benders’ decomposition approaches with application to chance-constrained problems with finite support

Author

Listed:
  • W. Ackooij

    (EDF R&D, OSIRIS)

  • A. Frangioni

    (Università di Pisa)

  • W. Oliveira

    (Universidade do Estado do Rio de Janeiro - UERJ)

Abstract

We explore modifications of the standard cutting-plane approach for minimizing a convex nondifferentiable function, given by an oracle, over a combinatorial set, which is the basis of the celebrated (generalized) Benders’ decomposition approach. Specifically, we combine stabilization—in two ways: via a trust region in the $$L_1$$ L 1 norm, or via a level constraint—and inexact function computation (solution of the subproblems). Managing both features simultaneously requires a nontrivial convergence analysis; we provide it under very weak assumptions on the handling of the two parameters (target and accuracy) controlling the informative on-demand inexact oracle corresponding to the subproblem, strengthening earlier know results. This yields new versions of Benders’ decomposition, whose numerical performance are assessed on a class of hybrid robust and chance-constrained problems that involve a random variable with an underlying discrete distribution, are convex in the decision variable, but have neither separable nor linear probabilistic constraints. The numerical results show that the approach has potential, especially for instances that are difficult to solve with standard techniques.

Suggested Citation

  • W. Ackooij & A. Frangioni & W. Oliveira, 2016. "Inexact stabilized Benders’ decomposition approaches with application to chance-constrained problems with finite support," Computational Optimization and Applications, Springer, vol. 65(3), pages 637-669, December.
    • Handle: RePEc:spr:coopap:v:65:y:2016:i:3:d:10.1007_s10589-016-9851-z
    DOI: 10.1007/s10589-016-9851-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-016-9851-z
    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/s10589-016-9851-z?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. Hanif Sherali & Brian Lunday, 2013. "On generating maximal nondominated Benders cuts," Annals of Operations Research, Springer, vol. 210(1), pages 57-72, November.
    2. Darinka Dentcheva & Bogumila Lai & Andrzej Ruszczyński, 2004. "Dual methods for probabilistic optimization problems ," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(2), pages 331-346, October.
    3. Jennifer Dinter & Steffen Rebennack & Josef Kallrath & Paul Denholm & Alexandra Newman, 2013. "The unit commitment model with concave emissions costs: a hybrid Benders’ Decomposition with nonconvex master problems," Annals of Operations Research, Springer, vol. 210(1), pages 361-386, November.
    4. de Klerk, E., 2006. "The Complexity of Optimizing over a Simplex, Hypercube or Sphere : A Short Survey," Discussion Paper 2006-85, Tilburg University, Center for Economic Research.
    5. Wim Ackooij & Jérôme Malick, 2016. "Decomposition algorithm for large-scale two-stage unit-commitment," Annals of Operations Research, Springer, vol. 238(1), pages 587-613, March.
    6. Wim Ackooij & Welington Oliveira, 2014. "Level bundle methods for constrained convex optimization with various oracles," Computational Optimization and Applications, Springer, vol. 57(3), pages 555-597, April.
    7. de Klerk, E., 2008. "The complexity of optimizing over a simplex, hypercube or sphere : A short survey," Other publications TiSEM 485b6860-cf1d-4cad-97b8-2, Tilburg University, School of Economics and Management.
    8. Santoso, Tjendera & Ahmed, Shabbir & Goetschalckx, Marc & Shapiro, Alexander, 2005. "A stochastic programming approach for supply chain network design under uncertainty," European Journal of Operational Research, Elsevier, vol. 167(1), pages 96-115, November.
    9. Wim Ackooij & Jérôme Malick, 2016. "Decomposition algorithm for large-scale two-stage unit-commitment," Annals of Operations Research, Springer, vol. 238(1), pages 587-613, March.
    10. Wolf, Christian & Fábián, Csaba I. & Koberstein, Achim & Suhl, Leena, 2014. "Applying oracles of on-demand accuracy in two-stage stochastic programming – A computational study," European Journal of Operational Research, Elsevier, vol. 239(2), pages 437-448.
    11. Etienne Klerk, 2008. "The complexity of optimizing over a simplex, hypercube or sphere: a short survey," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 111-125, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Shuai Liu, 2019. "A simple version of bundle method with linear programming," Computational Optimization and Applications, Springer, vol. 72(2), pages 391-412, March.
    2. Holger Berthold & Holger Heitsch & René Henrion & Jan Schwientek, 2022. "On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 1-37, August.
    3. T. González Grandón & H. Heitsch & R. Henrion, 2017. "A joint model of probabilistic/robust constraints for gas transport management in stationary networks," Computational Management Science, Springer, vol. 14(3), pages 443-460, July.
    4. Wim Ackooij & Welington Oliveira & Yongjia Song, 2019. "On level regularization with normal solutions in decomposition methods for multistage stochastic programming problems," Computational Optimization and Applications, Springer, vol. 74(1), pages 1-42, September.
    5. Lukáš Adam & Martin Branda & Holger Heitsch & René Henrion, 2020. "Solving joint chance constrained problems using regularization and Benders’ decomposition," Annals of Operations Research, Springer, vol. 292(2), pages 683-709, September.
    6. Daniel Baena & Jordi Castro & Antonio Frangioni, 2020. "Stabilized Benders Methods for Large-Scale Combinatorial Optimization, with Application to Data Privacy," Management Science, INFORMS, vol. 66(7), pages 3051-3068, July.
    7. Martina Kuchlbauer & Frauke Liers & Michael Stingl, 2022. "Outer Approximation for Mixed-Integer Nonlinear Robust Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 1056-1086, December.
    8. Xiao Liu & Simge Küçükyavuz, 2018. "A polyhedral study of the static probabilistic lot-sizing problem," Annals of Operations Research, Springer, vol. 261(1), pages 233-254, February.
    9. W. Ackooij & S. Demassey & P. Javal & H. Morais & W. Oliveira & B. Swaminathan, 2021. "A bundle method for nonsmooth DC programming with application to chance-constrained problems," Computational Optimization and Applications, Springer, vol. 78(2), pages 451-490, March.

    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. Wim van Ackooij & Welington de Oliveira & Yongjia Song, 2018. "Adaptive Partition-Based Level Decomposition Methods for Solving Two-Stage Stochastic Programs with Fixed Recourse," INFORMS Journal on Computing, INFORMS, vol. 30(1), pages 57-70, February.
    2. Wim Ackooij & Nicolas Lebbe & Jérôme Malick, 2017. "Regularized decomposition of large scale block-structured robust optimization problems," Computational Management Science, Springer, vol. 14(3), pages 393-421, July.
    3. Maziar Salahi, 2010. "Convex optimization approach to a single quadratically constrained quadratic minimization problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 18(2), pages 181-187, June.
    4. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2020. "An active-set algorithmic framework for non-convex optimization problems over the simplex," Computational Optimization and Applications, Springer, vol. 77(1), pages 57-89, September.
    5. Blanchot, Xavier & Clautiaux, François & Detienne, Boris & Froger, Aurélien & Ruiz, Manuel, 2023. "The Benders by batch algorithm: Design and stabilization of an enhanced algorithm to solve multicut Benders reformulation of two-stage stochastic programs," European Journal of Operational Research, Elsevier, vol. 309(1), pages 202-216.
    6. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
    7. Immanuel Bomze & Stefan Gollowitzer & E. Yıldırım, 2014. "Rounding on the standard simplex: regular grids for global optimization," Journal of Global Optimization, Springer, vol. 59(2), pages 243-258, July.
    8. Marco Locatelli, 2013. "Approximation algorithm for a class of global optimization problems," Journal of Global Optimization, Springer, vol. 55(1), pages 13-25, January.
    9. Alexander Franz & Julia Rieck & Jürgen Zimmermann, 2019. "Fix-and-optimize procedures for solving the long-term unit commitment problem with pumped storages," Annals of Operations Research, Springer, vol. 274(1), pages 241-265, March.
    10. Rahmaniani, Ragheb & Crainic, Teodor Gabriel & Gendreau, Michel & Rei, Walter, 2017. "The Benders decomposition algorithm: A literature review," European Journal of Operational Research, Elsevier, vol. 259(3), pages 801-817.
    11. Immanuel Bomze & Werner Schachinger & Gabriele Uchida, 2012. "Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 423-445, March.
    12. Immanuel M. Bomze & Werner Schachinger & Reinhard Ullrich, 2018. "The Complexity of Simple Models—A Study of Worst and Typical Hard Cases for the Standard Quadratic Optimization Problem," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 651-674, May.
    13. W. Ackooij & X. Warin, 2020. "On conditional cuts for stochastic dual dynamic programming," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(2), pages 173-199, June.
    14. Clara Lage & Claudia Sagastizábal & Mikhail Solodov, 2020. "Multiplier Stabilization Applied to Two-Stage Stochastic Programs," Documents de travail du Centre d'Economie de la Sorbonne 20010, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    15. Jeihoonian, Mohammad & Kazemi Zanjani, Masoumeh & Gendreau, Michel, 2016. "Accelerating Benders decomposition for closed-loop supply chain network design: Case of used durable products with different quality levels," European Journal of Operational Research, Elsevier, vol. 251(3), pages 830-845.
    16. Clara Lage & Claudia Sagastizábal & Mikhail Solodov, 2019. "Multiplier Stabilization Applied to Two-Stage Stochastic Programs," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 158-178, October.
    17. Clara Lage & Claudia Sagastizábal & Mikhail Solodov, 2020. "Multiplier Stabilization Applied to Two-Stage Stochastic Programs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02900862, HAL.
    18. Keyvanshokooh, Esmaeil & Ryan, Sarah M. & Kabir, Elnaz, 2016. "Hybrid robust and stochastic optimization for closed-loop supply chain network design using accelerated Benders decomposition," European Journal of Operational Research, Elsevier, vol. 249(1), pages 76-92.
    19. M. Jenabi & S. M. T. Fatemi Ghomi & S. A. Torabi & Moeen Sammak Jalali, 2022. "An accelerated Benders decomposition algorithm for stochastic power system expansion planning using sample average approximation," OPSEARCH, Springer;Operational Research Society of India, vol. 59(4), pages 1304-1336, December.
    20. de Klerk, E. & Laurent, M., 2010. "Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube," Other publications TiSEM 619d9658-77df-4b5e-9868-0, Tilburg University, School of Economics and Management.

    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:coopap:v:65:y:2016:i:3:d:10.1007_s10589-016-9851-z. 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.