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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
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    References listed on IDEAS

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    1. 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.
    2. George B. Dantzig, 1990. "The Diet Problem," Interfaces, INFORMS, vol. 20(4), pages 43-47, August.
    3. Miles Lubin & Iain Dunning, 2015. "Computing in Operations Research Using Julia," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 238-248, May.
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    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.

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