IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v315y2022i1d10.1007_s10479-022-04698-0.html
   My bibliography  Save this article

Limit laws for empirical optimal solutions in random linear programs

Author

Listed:
  • Marcel Klatt

    (University of Göttingen)

  • Axel Munk

    (University of Göttingen
    Max Planck Institute for Biophysical Chemistry)

  • Yoav Zemel

    (University of Cambridge)

Abstract

We consider a general linear program in standard form whose right-hand side constraint vector is subject to random perturbations. For the corresponding random linear program, we characterize under general assumptions the random fluctuations of the empirical optimal solutions around their population quantities after standardization by a distributional limit theorem. Our approach is geometric in nature and further relies on duality and the collection of dual feasible basic solutions. The limiting random variables are driven by the amount of degeneracy inherent in linear programming. In particular, if the corresponding dual linear program is degenerate the asymptotic limit law might not be unique and is determined from the way the empirical optimal solution is chosen. Furthermore, we include consistency and convergence rates of the Hausdorff distance between the empirical and the true optimality sets as well as a limit law for the empirical optimal value involving the set of all dual optimal basic solutions. Our analysis is motivated from statistical optimal transport that is of particular interest here and distributional limit laws for empirical optimal transport plans follow by a simple application of our general theory. The corresponding limit distribution is usually non-Gaussian which stands in strong contrast to recent finding for empirical entropy regularized optimal transport solutions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:315:y:2022:i:1:d:10.1007_s10479-022-04698-0
    DOI: 10.1007/s10479-022-04698-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-022-04698-0
    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/s10479-022-04698-0?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. repec:hal:spmain:info:hdl:2441/64itsev5509q8aa5mrbhi0g0b6 is not listed on IDEAS
    2. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    3. Alan J. King & R. Tyrrell Rockafellar, 1993. "Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 148-162, February.
    4. J. T. Chang & D. Pollard, 1997. "Conditioning as disintegration," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 51(3), pages 287-317, November.
    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. Harvey J. Greenberg, 1986. "An analysis of degeneracy," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 33(4), pages 635-655, November.
    7. Allen R. Ferguson & George B. Dantzig, 1956. "The Allocation of Aircraft to Routes--An Example of Linear Programming Under Uncertain Demand," Management Science, INFORMS, vol. 3(1), pages 45-73, October.
    8. Alexander Shapiro, 1993. "Asymptotic Behavior of Optimal Solutions in Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 829-845, November.
    9. Hadigheh, Alireza Ghaffari & Terlaky, Tamas, 2006. "Sensitivity analysis in linear optimization: Invariant support set intervals," European Journal of Operational Research, Elsevier, vol. 169(3), pages 1158-1175, March.
    10. 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.
    11. George B. Dantzig, 1955. "Linear Programming under Uncertainty," Management Science, INFORMS, vol. 1(3-4), pages 197-206, 04-07.
    12. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    13. Alan J. King, 1989. "Generalized Delta Theorems for Multivalued Mappings and Measurable Selections," Mathematics of Operations Research, INFORMS, vol. 14(4), pages 720-736, November.
    14. Stephen M. Robinson, 1977. "A Characterization of Stability in Linear Programming," Operations Research, INFORMS, vol. 25(3), pages 435-447, 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. Gunsilius, Florian F., 2023. "A condition for the identification of multivariate models with binary instruments," Journal of Econometrics, Elsevier, vol. 235(1), pages 220-238.
    2. G. Guerkan & A.Y. Oezge & S.M. Robinson, 1994. "Sample-Path Optimization in Simulation," Working Papers wp94070, International Institute for Applied Systems Analysis.
    3. Baha Alzalg & Asma Gafour, 2023. "Convergence of a Weighted Barrier Algorithm for Stochastic Convex Quadratic Semidefinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 490-515, February.
    4. Hongjian Shi & Mathias Drton & Marc Hallin & Fang Han, 2023. "Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics," Working Papers ECARES 2023-03, ULB -- Universite Libre de Bruxelles.
    5. Sanjay Mehrotra & M. Gokhan Ozevin, 2009. "Decomposition Based Interior Point Methods for Two-Stage Stochastic Convex Quadratic Programs with Recourse," Operations Research, INFORMS, vol. 57(4), pages 964-974, August.
    6. Richard W. Cottle, 2005. "George B. Dantzig: Operations Research Icon," Operations Research, INFORMS, vol. 53(6), pages 892-898, December.
    7. Alberto González-Sanz & Marc Hallin & Bodhisattva Sen, 2023. "Monotone Measure-Preserving Maps in Hilbert Spaces: Existence, Uniqueness, and Stability," Working Papers ECARES 2023-10, ULB -- Universite Libre de Bruxelles.
    8. Silvia Vogel, 2006. "Semiconvergence in distribution of random closed sets with application to random optimization problems," Annals of Operations Research, Springer, vol. 142(1), pages 269-282, February.
    9. Olivier Paul Faugeras & Ludger Rüschendorf, 2021. "Functional, randomized and smoothed multivariate quantile regions," Post-Print hal-03352330, HAL.
    10. 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.
    11. Rebecca Stockbridge & Güzin Bayraksan, 2016. "Variance reduction in Monte Carlo sampling-based optimality gap estimators for two-stage stochastic linear programming," Computational Optimization and Applications, Springer, vol. 64(2), pages 407-431, June.
    12. Giorgio & Cesare, 2018. "A Tutorial on Sensitivity and Stability in Nonlinear Programming and Variational Inequalities under Differentiability Assumptions," DEM Working Papers Series 159, University of Pavia, Department of Economics and Management.
    13. T. Glenn Bailey & Paul A. Jensen & David P. Morton, 1999. "Response surface analysis of two‐stage stochastic linear programming with recourse," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(7), pages 753-776, October.
    14. M. J. Cánovas & R. Henrion & M. A. López & J. Parra, 2016. "Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 925-952, June.
    15. Fan, Yanqin & Henry, Marc, 2023. "Vector copulas," Journal of Econometrics, Elsevier, vol. 234(1), pages 128-150.
    16. Warren B. Powell, 2016. "Perspectives of approximate dynamic programming," Annals of Operations Research, Springer, vol. 241(1), pages 319-356, June.
    17. Dupacova, Jitka, 2002. "Applications of stochastic programming: Achievements and questions," European Journal of Operational Research, Elsevier, vol. 140(2), pages 281-290, July.
    18. Garcia, Diego, 2003. "Convergence and Biases of Monte Carlo estimates of American option prices using a parametric exercise rule," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1855-1879, August.
    19. Hsieh, Yu-Wei & Shi, Xiaoxia & Shum, Matthew, 2022. "Inference on estimators defined by mathematical programming," Journal of Econometrics, Elsevier, vol. 226(2), pages 248-268.
    20. R.J.-B. Wets, 1994. "Challenges in Stochastic Programming," Working Papers wp94032, International Institute for Applied Systems Analysis.

    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:annopr:v:315:y:2022:i:1:d:10.1007_s10479-022-04698-0. 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.