IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v91y2025i2d10.1007_s10898-024-01370-8.html
   My bibliography  Save this article

Global solution of quadratic problems using interval methods and convex relaxations

Author

Listed:
  • Sourour Elloumi

    (Institut Polytechnique de Paris
    Cnam)

  • Amélie Lambert

    (Cnam)

  • Bertrand Neveu

    (Univ. Gustave Eiffel)

  • Gilles Trombettoni

    (University of Montpellier)

Abstract

Interval branch-and-bound solvers provide reliable algorithms for handling non-convex optimization problems by ensuring the feasibility and the optimality of the computed solutions, i.e. independently from the floating-point rounding errors. Moreover, these solvers deal with a wide variety of mathematical operators. However, these solvers are not dedicated to quadratic optimization and do not exploit nonlinear convex relaxations in their framework. We present an interval branch-and-bound method that can efficiently solve quadratic optimization problems. At each node explored by the algorithm, our solver uses a quadratic convex relaxation which is as strong as a semi-definite programming relaxation, and a variable selection strategy dedicated to quadratic problems. The interval features can then propagate efficiently this information for contracting all variable domains. We also propose to make our algorithm rigorous by certifying firstly the convexity of the objective function of our relaxation, and secondly the validity of the lower bound calculated at each node. In the non-rigorous case, our experiments show significant speedups on general integer quadratic instances, and when reliability is required, our first results show that we are able to handle medium-sized instances in a reasonable running time.

Suggested Citation

  • Sourour Elloumi & Amélie Lambert & Bertrand Neveu & Gilles Trombettoni, 2025. "Global solution of quadratic problems using interval methods and convex relaxations," Journal of Global Optimization, Springer, vol. 91(2), pages 331-353, February.
  • Handle: RePEc:spr:jglopt:v:91:y:2025:i:2:d:10.1007_s10898-024-01370-8
    DOI: 10.1007/s10898-024-01370-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-024-01370-8
    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/s10898-024-01370-8?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. Hering, Dominik & Xhonneux, André & Müller, Dirk, 2021. "Design optimization of a heating network with multiple heat pumps using mixed integer quadratically constrained programming," Energy, Elsevier, vol. 226(C).
    2. Bertrand Neveu & Gilles Trombettoni & Ignacio Araya, 2016. "Node selection strategies in interval Branch and Bound algorithms," Journal of Global Optimization, Springer, vol. 64(2), pages 289-304, February.
    3. Ignacio Araya & Gilles Trombettoni & Bertrand Neveu & Gilles Chabert, 2014. "Upper bounding in inner regions for global optimization under inequality constraints," Journal of Global Optimization, Springer, vol. 60(2), pages 145-164, October.
    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. Bertrand Neveu & Martin Gorce & Pascal Monasse & Gilles Trombettoni, 2019. "A generic interval branch and bound algorithm for parameter estimation," Journal of Global Optimization, Springer, vol. 73(3), pages 515-535, March.
    2. Ignacio Araya & Bertrand Neveu, 2018. "lsmear: a variable selection strategy for interval branch and bound solvers," Journal of Global Optimization, Springer, vol. 71(3), pages 483-500, July.
    3. Ignacio Araya & Jose Campusano & Damir Aliquintui, 2019. "Nonlinear biobjective optimization: improvements to interval branch & bound algorithms," Journal of Global Optimization, Springer, vol. 75(1), pages 91-110, September.
    4. Victor Reyes & Ignacio Araya, 2025. "Node selection through upper bounding local search methods in branch & bound solvers for NCOPs," Journal of Global Optimization, Springer, vol. 91(2), pages 355-369, February.
    5. Yao, Shuai & Wu, Jianzhong & Qadrdan, Meysam, 2024. "A state-of-the-art analysis and perspectives on the 4th/5th generation district heating and cooling systems," Renewable and Sustainable Energy Reviews, Elsevier, vol. 202(C).
    6. Victor Reyes & Ignacio Araya, 2021. "AbsTaylor: upper bounding with inner regions in nonlinear continuous global optimization problems," Journal of Global Optimization, Springer, vol. 79(2), pages 413-429, February.
    7. Ignacio Araya & Damir Aliquintui & Franco Ardiles & Braulio Lobo, 2021. "Nonlinear biobjective optimization: improving the upper envelope using feasible line segments," Journal of Global Optimization, Springer, vol. 79(2), pages 503-520, February.
    8. Ignacio Araya & Victor Reyes, 2016. "Interval Branch-and-Bound algorithms for optimization and constraint satisfaction: a survey and prospects," Journal of Global Optimization, Springer, vol. 65(4), pages 837-866, August.
    9. Stock, Jan & Schmidt, Till & Xhonneux, André & Müller, Dirk, 2024. "Optimisation of district heating transformation for the efficient integration of a low-temperature heat source," Energy, Elsevier, vol. 308(C).
    10. Hering, Dominik & Faller, Michael R. & Xhonneux, André & Müller, Dirk, 2022. "Operational optimization of a 4th generation district heating network with mixed integer quadratically constrained programming," Energy, Elsevier, vol. 250(C).
    11. Stock, Jan & Xhonneux, André & Müller, Dirk, 2024. "Optimisation of district heating network separation for the utilisation of heat source potentials," Energy, Elsevier, vol. 303(C).
    12. Zheng, Xuejing & Shi, Zhiyuan & Wang, Yaran & Zhang, Huan & Liu, Huzhen, 2023. "Thermo-hydraulic condition optimization of large-scale complex district heating network: A case study of Tianjin," Energy, Elsevier, vol. 266(C).
    13. Ignacio Araya & Frédéric Messine & Jordan Ninin & Gilles Trombettoni, 2025. "Hybridizing two linear relaxation techniques in interval-based solvers," Journal of Global Optimization, Springer, vol. 91(3), pages 437-456, March.
    14. Bertrand Neveu & Gilles Trombettoni & Ignacio Araya, 2016. "Node selection strategies in interval Branch and Bound algorithms," Journal of Global Optimization, Springer, vol. 64(2), pages 289-304, February.
    15. Marendet, Antoine & Goldsztejn, Alexandre & Chabert, Gilles & Jermann, Christophe, 2020. "A standard branch-and-bound approach for nonlinear semi-infinite problems," European Journal of Operational Research, Elsevier, vol. 282(2), pages 438-452.

    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:jglopt:v:91:y:2025:i:2:d:10.1007_s10898-024-01370-8. 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.