IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v43y2018i4p1051-1071.html
   My bibliography  Save this article

Random Projections for Linear Programming

Author

Listed:
  • Ky Vu

    (Institute of Theoretical Computer Science and Communications (ITCSC), Chinese University of Hong Kong, Hong Kong, China)

  • Pierre-Louis Poirion

    (Huawei Research Center, Paris, France)

  • Leo Liberti

    (National Center for Scientific Research (CNRS) LIX, Ecole Polytechnique, 91128 Palaiseau, France)

Abstract

Random projections are random linear maps, sampled from appropriate distributions, which approximately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well known Johnson-Lindenstrauss lemma states that there are random matrices with surprisingly few rows which approximately preserve pairwise Euclidean distances among a set of points. This is commonly used to speed up algorithms based on Euclidean distances. We prove that these matrices also preserve other quantities, such as the distance to a cone. We exploit this result to devise a probabilistic algorithm to approximately solve linear programs. We show that this algorithm can approximately solve very large randomly generated LP instances. We also showcase its application to an error correction coding problem.

Suggested Citation

  • Ky Vu & Pierre-Louis Poirion & Leo Liberti, 2018. "Random Projections for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1051-1071, November.
    • Handle: RePEc:inm:ormoor:v:43:y:2018:i:4:p:1051-1071
    DOI: 10.1287/moor.2017.0894
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/moor.2017.0894
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2017.0894?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
    ---><---

    References listed on IDEAS

    as
    1. George B. Dantzig, 1990. "The Diet Problem," Interfaces, INFORMS, vol. 20(4), pages 43-47, August.
    2. Miles Lubin & Iain Dunning, 2015. "Computing in Operations Research Using Julia," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 238-248, May.
    3. Daniela Pucci de Farias & Benjamin Van Roy, 2004. "On Constraint Sampling in the Linear Programming Approach to Approximate Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 462-478, August.
    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. Leo Liberti & Benedetto Manca, 2022. "Side-constrained minimum sum-of-squares clustering: mathematical programming and random projections," Journal of Global Optimization, Springer, vol. 83(1), pages 83-118, May.
    2. Leo Liberti, 2020. "Distance geometry and data science," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 271-339, July.

    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. Meissner, Joern & Strauss, Arne, 2012. "Network revenue management with inventory-sensitive bid prices and customer choice," European Journal of Operational Research, Elsevier, vol. 216(2), pages 459-468.
    2. F Dubeau, 2019. "Behavior Related to Taxation System: Example of Bi-Criteria Linear Program for Animal Diet Formulation," Annals of Social Sciences & Management studies, Juniper Publishers Inc., vol. 3(3), pages 60-65, April.
    3. Aritra Pal & Hadi Charkhgard, 2019. "A Feasibility Pump and Local Search Based Heuristic for Bi-Objective Pure Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 31(1), pages 115-133, February.
    4. Amir Ali Nasrollahzadeh & Amin Khademi & Maria E. Mayorga, 2018. "Real-Time Ambulance Dispatching and Relocation," Manufacturing & Service Operations Management, INFORMS, vol. 20(3), pages 467-480, July.
    5. Thomas W. M. Vossen & Dan Zhang, 2015. "Reductions of Approximate Linear Programs for Network Revenue Management," Operations Research, INFORMS, vol. 63(6), pages 1352-1371, December.
    6. Dimitris Bertsimas & Arthur Delarue & Patrick Jaillet & Sébastien Martin, 2019. "Travel Time Estimation in the Age of Big Data," Operations Research, INFORMS, vol. 67(2), pages 498-515, March.
    7. Torres-Rojo, J. M., 2001. "Risk management in the design of a feeding ration: a portfolio theory approach," Agricultural Systems, Elsevier, vol. 68(1), pages 1-20, April.
    8. Cao, Yankai & Zavala, Victor M. & D’Amato, Fernando, 2018. "Using stochastic programming and statistical extrapolation to mitigate long-term extreme loads in wind turbines," Applied Energy, Elsevier, vol. 230(C), pages 1230-1241.
    9. Takafumi Kanamori & Akiko Takeda, 2012. "Worst-Case Violation of Sampled Convex Programs for Optimization with Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 152(1), pages 171-197, January.
    10. Eli Towle & James Luedtke, 2018. "New solution approaches for the maximum-reliability stochastic network interdiction problem," Computational Management Science, Springer, vol. 15(3), pages 455-477, October.
    11. Diego Klabjan & Daniel Adelman, 2007. "An Infinite-Dimensional Linear Programming Algorithm for Deterministic Semi-Markov Decision Processes on Borel Spaces," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 528-550, August.
    12. L. Jeff Hong & Zhiyuan Huang & Henry Lam, 2021. "Learning-Based Robust Optimization: Procedures and Statistical Guarantees," Management Science, INFORMS, vol. 67(6), pages 3447-3467, June.
    13. Guanglei Wang & Hassan Hijazi, 2018. "Mathematical programming methods for microgrid design and operations: a survey on deterministic and stochastic approaches," Computational Optimization and Applications, Springer, vol. 71(2), pages 553-608, November.
    14. Nemirovski, Arkadi, 2012. "On safe tractable approximations of chance constraints," European Journal of Operational Research, Elsevier, vol. 219(3), pages 707-718.
    15. Davi Valladão & Thuener Silva & Marcus Poggi, 2019. "Time-consistent risk-constrained dynamic portfolio optimization with transactional costs and time-dependent returns," Annals of Operations Research, Springer, vol. 282(1), pages 379-405, November.
    16. Onur Babat & Juan C. Vera & Luis F. Zuluaga, 2021. "Computing near-optimal Value-at-Risk portfolios using Integer Programming techniques," Papers 2107.07339, arXiv.org.
    17. Jens Weibezahn & Mario Kendziorski, 2019. "Illustrating the Benefits of Openness: A Large-Scale Spatial Economic Dispatch Model Using the Julia Language," Energies, MDPI, vol. 12(6), pages 1-21, March.
    18. François Dubeau & Pierre-Olivier Julien & Candido Pomar, 2011. "Formulating diets for growing pigs: economic and environmental considerations," Annals of Operations Research, Springer, vol. 190(1), pages 239-269, October.
    19. M. C. Campi & S. Garatti, 2011. "A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 257-280, February.
    20. Dimitris Bertsimas & Patrick Jaillet, & Sébastien Martin, 2019. "Online Vehicle Routing: The Edge of Optimization in Large-Scale Applications," Operations Research, INFORMS, vol. 67(1), pages 143-162, January.

    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:inm:ormoor:v:43:y:2018:i:4:p:1051-1071. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.