IDEAS home Printed from https://ideas.repec.org/a/spr/comgts/v14y2017i4d10.1007_s10287-017-0290-9.html
   My bibliography  Save this article

Centered solutions for uncertain linear equations

Author

Listed:
  • Jianzhe Zhen

    (Tilburg University)

  • Dick Hertog

    (Tilburg University)

Abstract

Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we derive convex representations for united and tolerable solution sets. Secondly, to obtain centered solutions for uncertain linear equations, we develop a new method based on adjustable robust optimization (ARO) techniques to compute the maximum size inscribed convex body (MCB) of the set of the solutions. In general, the obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We use recent results from ARO to characterize for which convex bodies the obtained MCB is optimal. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input–output model, Colley’s Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.

Suggested Citation

  • Jianzhe Zhen & Dick Hertog, 2017. "Centered solutions for uncertain linear equations," Computational Management Science, Springer, vol. 14(4), pages 585-610, October.
    • Handle: RePEc:spr:comgts:v:14:y:2017:i:4:d:10.1007_s10287-017-0290-9
    DOI: 10.1007/s10287-017-0290-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10287-017-0290-9
    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/s10287-017-0290-9?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. Robert L. Smith, 1984. "Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions," Operations Research, INFORMS, vol. 32(6), pages 1296-1308, December.
    2. Hladík, Milan & Popova, Evgenija D., 2015. "Maximal inner boxes in parametric AE-solution sets with linear shape," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 606-619.
    3. Zhen, J. & den Hertog, D., 2015. "Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection," Discussion Paper 2015-004, Tilburg University, Center for Economic Research.
    4. Zhen, J. & den Hertog, D., 2015. "Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection," Other publications TiSEM 4a51526e-c7f0-436f-aeaa-f, Tilburg University, School of Economics and Management.
    5. Blanc, J.P.C. & den Hertog, D., 2008. "On Markov Chains with Uncertain Data," Discussion Paper 2008-50, Tilburg University, Center for Economic Research.
    6. Bram L. Gorissen & Hans Blanc & Dick den Hertog & Aharon Ben-Tal, 2014. "Technical Note---Deriving Robust and Globalized Robust Solutions of Uncertain Linear Programs with General Convex Uncertainty Sets," Operations Research, INFORMS, vol. 62(3), pages 672-679, June.
    7. Burer Samuel, 2012. "Robust Rankings for College Football," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 8(2), pages 1-22, June.
    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. Zhen, Jianzhe & den Hertog, Dick, 2015. "Robust Solutions for Systems of Uncertain Linear Equations," Discussion Paper 2015-044, Tilburg University, Center for Economic Research.
    2. Zhen, Jianzhe & den Hertog, Dick, 2016. "Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044)," Other publications TiSEM 297fa3b1-5290-48b5-bbc0-0, Tilburg University, School of Economics and Management.
    3. Zhen, Jianzhe & den Hertog, Dick, 2015. "Robust Solutions for Systems of Uncertain Linear Equations," Other publications TiSEM d072bdb9-4168-4522-90d8-1, Tilburg University, School of Economics and Management.
    4. Zhen, Jianzhe & den Hertog, Dick, 2016. "Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044)," Discussion Paper 2016-048, Tilburg University, Center for Economic Research.
    5. Luca Anzilli & Silvio Giove, 2020. "Multi-criteria and medical diagnosis for application to health insurance systems: a general approach through non-additive measures," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(2), pages 559-582, December.
    6. Corrente, Salvatore & Figueira, José Rui & Greco, Salvatore, 2014. "The SMAA-PROMETHEE method," European Journal of Operational Research, Elsevier, vol. 239(2), pages 514-522.
    7. Stephen Baumert & Archis Ghate & Seksan Kiatsupaibul & Yanfang Shen & Robert L. Smith & Zelda B. Zabinsky, 2009. "Discrete Hit-and-Run for Sampling Points from Arbitrary Distributions Over Subsets of Integer Hyperrectangles," Operations Research, INFORMS, vol. 57(3), pages 727-739, June.
    8. Villacorta, Pablo J. & Verdegay, José L., 2016. "FuzzyStatProb: An R Package for the Estimation of Fuzzy Stationary Probabilities from a Sequence of Observations of an Unknown Markov Chain," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 71(i08).
    9. Qi Fan & Jiaqiao Hu, 2018. "Surrogate-Based Promising Area Search for Lipschitz Continuous Simulation Optimization," INFORMS Journal on Computing, INFORMS, vol. 30(4), pages 677-693, November.
    10. de Ruiter, Frans, 2018. "Primal and dual approaches to adjustable robust optimization," Other publications TiSEM 4ce224f3-d9fc-4283-b5a6-4, Tilburg University, School of Economics and Management.
    11. Reuven Rubinstein, 2009. "The Gibbs Cloner for Combinatorial Optimization, Counting and Sampling," Methodology and Computing in Applied Probability, Springer, vol. 11(4), pages 491-549, December.
    12. Luis V. Montiel & J. Eric Bickel, 2014. "A Generalized Sampling Approach for Multilinear Utility Functions Given Partial Preference Information," Decision Analysis, INFORMS, vol. 11(3), pages 147-170, September.
    13. Pavel Shcherbakov & Mingyue Ding & Ming Yuchi, 2021. "Random Sampling Many-Dimensional Sets Arising in Control," Mathematics, MDPI, vol. 9(5), pages 1-16, March.
    14. Cheng, Haiyan & Sandu, Adrian, 2009. "Efficient uncertainty quantification with the polynomial chaos method for stiff systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(11), pages 3278-3295.
    15. Sorawit Saengkyongam & Anthony Hayter & Seksan Kiatsupaibul & Wei Liu, 2020. "Efficient computation of the stochastic behavior of partial sum processes," Computational Statistics, Springer, vol. 35(1), pages 343-358, March.
    16. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2020. "Robustness Characterizations for Uncertain Optimization Problems via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 459-479, August.
    17. Clive B Beggs & Alexander J Bond & Stacey Emmonds & Ben Jones, 2019. "Hidden dynamics of soccer leagues: The predictive ‘power’ of partial standings," PLOS ONE, Public Library of Science, vol. 14(12), pages 1-28, December.
    18. Ru, Zice & Liu, Jiapeng & Kadziński, Miłosz & Liao, Xiuwu, 2023. "Probabilistic ordinal regression methods for multiple criteria sorting admitting certain and uncertain preferences," European Journal of Operational Research, Elsevier, vol. 311(2), pages 596-616.
    19. Jan Heufer, 2014. "Generating Random Optimising Choices," Computational Economics, Springer;Society for Computational Economics, vol. 44(3), pages 295-305, October.
    20. Bram L. Gorissen & Hans Blanc & Dick den Hertog & Aharon Ben-Tal, 2014. "Technical Note---Deriving Robust and Globalized Robust Solutions of Uncertain Linear Programs with General Convex Uncertainty Sets," Operations Research, INFORMS, vol. 62(3), pages 672-679, June.

    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:comgts:v:14:y:2017:i:4:d:10.1007_s10287-017-0290-9. 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.