IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v92y2020i1d10.1007_s00186-020-00703-z.html
   My bibliography  Save this article

Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case

Author

Listed:
  • Valentin Hartmann

    (University of Goettingen
    EPFL)

  • Dominic Schuhmacher

    (University of Goettingen)

Abstract

We consider the problem of finding an optimal transport plan between an absolutely continuous measure and a finitely supported measure of the same total mass when the transport cost is the unsquared Euclidean distance. We may think of this problem as closest distance allocation of some resource continuously distributed over Euclidean space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost. We present an algorithm for computing the optimal transport plan, which is similar to the approach for the squared Euclidean cost by Aurenhammer et al. (Algorithmica 20(1):61–76, 1998) and Mérigot (Comput Graph Forum 30(5):1583–1592, 2011). We show the necessary results to make the approach work for the Euclidean cost, evaluate its performance on a set of test cases, and give a number of applications. The later include goodness-of-fit partitions, a novel visual tool for assessing whether a finite sample is consistent with a posited probability density.

Suggested Citation

  • Valentin Hartmann & Dominic Schuhmacher, 2020. "Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 133-163, August.
    • Handle: RePEc:spr:mathme:v:92:y:2020:i:1:d:10.1007_s00186-020-00703-z
    DOI: 10.1007/s00186-020-00703-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-020-00703-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/s00186-020-00703-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. Sándor P. Fekete & Joseph S. B. Mitchell & Karin Beurer, 2005. "On the Continuous Fermat-Weber Problem," Operations Research, INFORMS, vol. 53(1), pages 61-76, February.
    2. Leon Cooper, 1972. "The Transportation-Location Problem," Operations Research, INFORMS, vol. 20(1), pages 94-108, February.
    3. Hanif D. Sherali & Frederick L. Nordai, 1988. "NP-Hard, Capacitated, Balanced p -Median Problems on a Chain Graph with a Continuum of Link Demands," Mathematics of Operations Research, INFORMS, vol. 13(1), pages 32-49, February.
    4. David G. Luenberger & Yinyu Ye, 2008. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 0, number 978-0-387-74503-9, April.
    5. Max Sommerfeld & Axel Munk, 2018. "Inference for empirical Wasserstein distances on finite spaces," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(1), pages 219-238, January.
    6. Lina Mallozzi & Justo Puerto & Moisés Rodríguez-Madrena, 2019. "On Location-Allocation Problems for Dimensional Facilities," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 730-767, August.
    7. Rippl, Thomas & Munk, Axel & Sturm, Anja, 2016. "Limit laws of the empirical Wasserstein distance: Gaussian distributions," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 90-109.
    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. Blanco, Víctor & Gázquez, Ricardo & Ponce, Diego & Puerto, Justo, 2023. "A branch-and-price approach for the continuous multifacility monotone ordered median problem," European Journal of Operational Research, Elsevier, vol. 306(1), pages 105-126.
    2. N Aras & M Orbay & I K Altinel, 2008. "Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(1), pages 64-79, January.
    3. Necati Aras & İ. Kuban Altınel & Metin Orbay, 2007. "New heuristic methods for the capacitated multi‐facility Weber problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 54(1), pages 21-32, February.
    4. Chandra Ade Irawan & Said Salhi & Kusmaningrum Soemadi, 2020. "The continuous single-source capacitated multi-facility Weber problem with setup costs: formulation and solution methods," Journal of Global Optimization, Springer, vol. 78(2), pages 271-294, October.
    5. M. Akyüz & İ. Altınel & Temel Öncan, 2014. "Location and allocation based branch and bound algorithms for the capacitated multi-facility Weber problem," Annals of Operations Research, Springer, vol. 222(1), pages 45-71, November.
    6. Chandra Ade Irawan & Martino Luis & Said Salhi & Arif Imran, 2019. "The incorporation of fixed cost and multilevel capacities into the discrete and continuous single source capacitated facility location problem," Annals of Operations Research, Springer, vol. 275(2), pages 367-392, April.
    7. Marcel Klatt & Axel Munk & Yoav Zemel, 2022. "Limit laws for empirical optimal solutions in random linear programs," Annals of Operations Research, Springer, vol. 315(1), pages 251-278, August.
    8. Cristiana L. Lara & Francisco Trespalacios & Ignacio E. Grossmann, 2018. "Global optimization algorithm for capacitated multi-facility continuous location-allocation problems," Journal of Global Optimization, Springer, vol. 71(4), pages 871-889, August.
    9. Shahmohammadi, Ali & Sioshansi, Ramteen & Conejo, Antonio J. & Afsharnia, Saeed, 2018. "Market equilibria and interactions between strategic generation, wind, and storage," Applied Energy, Elsevier, vol. 220(C), pages 876-892.
    10. Alp Atakan & Mehmet Ekmekci & Ludovic Renou, 2021. "Cross-verification and Persuasive Cheap Talk," Papers 2102.13562, arXiv.org, revised Apr 2021.
    11. Arthur Medeiros & Thales Ramos & José Tavares de Oliveira & Manoel F. Medeiros Júnior, 2020. "Direct Voltage Control of a Doubly Fed Induction Generator by Means of Optimal Strategy," Energies, MDPI, vol. 13(3), pages 1-28, February.
    12. Ivorra, Benjamin & Mohammadi, Bijan & Manuel Ramos, Angel, 2015. "A multi-layer line search method to improve the initialization of optimization algorithms," European Journal of Operational Research, Elsevier, vol. 247(3), pages 711-720.
    13. Tanaka, Ken'ichiro & Toda, Alexis Akira, 2015. "Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis," University of California at San Diego, Economics Working Paper Series qt7g23r5kh, Department of Economics, UC San Diego.
    14. Daoqin Tong & Alan T. Murray, 2009. "Maximising coverage of spatial demand for service," Papers in Regional Science, Wiley Blackwell, vol. 88(1), pages 85-97, March.
    15. Ashrafi, M. & Khanjani, M.J. & Fadaei-Kermani, E. & Barani, G.A., 2015. "Farm drainage channel network optimization by improved modified minimal spanning tree," Agricultural Water Management, Elsevier, vol. 161(C), pages 1-8.
    16. Thomas Byrne & Sándor P. Fekete & Jörg Kalcsics & Linda Kleist, 2023. "Competitive location problems: balanced facility location and the One-Round Manhattan Voronoi Game," Annals of Operations Research, Springer, vol. 321(1), pages 79-101, February.
    17. Sergey Badikov & Antoine Jacquier & Daphne Qing Liu & Patrick Roome, 2016. "No-arbitrage bounds for the forward smile given marginals," Papers 1603.06389, arXiv.org, revised Oct 2016.
    18. Szidarovszky, Ferenc & Luo, Yi, 2014. "Incorporating risk seeking attitude into defense strategy," Reliability Engineering and System Safety, Elsevier, vol. 123(C), pages 104-109.
    19. Viet Anh Nguyen & Daniel Kuhn & Peyman Mohajerin Esfahani, 2018. "Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator," Papers 1805.07194, arXiv.org.
    20. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department 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:mathme:v:92:y:2020:i:1:d:10.1007_s00186-020-00703-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.