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Parallel distributed-memory simplex for large-scale stochastic LP problems

Author

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  • Miles Lubin
  • J. Hall
  • Cosmin Petra
  • Mihai Anitescu

Abstract

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hot-starts (e.g., in a branch-and-bound context) and can provide important sensitivity information. Our approach exploits the dual block-angular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for high-performance distributed-memory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual block-angular structure. Our implementation is competitive in serial with highly efficient sparsity-exploiting simplex codes and achieves significant relative speed-ups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largest-scale parallel sparsity-exploiting revised simplex implementation that has been developed to date and the first truly distributed solver. It is built on novel analysis of the linear algebra for dual block-angular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying product-form updates. Copyright Springer Science+Business Media New York (outside the USA) 2013

Suggested Citation

  • Miles Lubin & J. Hall & Cosmin Petra & Mihai Anitescu, 2013. "Parallel distributed-memory simplex for large-scale stochastic LP problems," Computational Optimization and Applications, Springer, vol. 55(3), pages 571-596, July.
  • Handle: RePEc:spr:coopap:v:55:y:2013:i:3:p:571-596
    DOI: 10.1007/s10589-013-9542-y
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    References listed on IDEAS

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    1. Harry M. Markowitz, 1957. "The Elimination form of the Inverse and its Application to Linear Programming," Management Science, INFORMS, vol. 3(3), pages 255-269, April.
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    4. Uwe H. Suhl & Leena M. Suhl, 1990. "Computing Sparse LU Factorizations for Large-Scale Linear Programming Bases," INFORMS Journal on Computing, INFORMS, vol. 2(4), pages 325-335, November.
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    Cited by:

    1. Qi Huangfu & J. Hall, 2015. "Novel update techniques for the revised simplex method," Computational Optimization and Applications, Springer, vol. 60(3), pages 587-608, April.
    2. Kuang-Yu Ding & Xin-Yee Lam & Kim-Chuan Toh, 2023. "On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems," Computational Optimization and Applications, Springer, vol. 86(1), pages 117-161, September.
    3. Martin Biel & Mikael Johansson, 2022. "Efficient Stochastic Programming in Julia," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 1885-1902, July.
    4. Lluís-Miquel Munguía & Geoffrey Oxberry & Deepak Rajan & Yuji Shinano, 2019. "Parallel PIPS-SBB: multi-level parallelism for stochastic mixed-integer programs," Computational Optimization and Applications, Springer, vol. 73(2), pages 575-601, June.
    5. ARAVENA, Ignacio & PAPAVASILIOU, Anthony, 2016. "An Asynchronous Distributed Algorithm for solving Stochastic Unit Commitment," LIDAM Discussion Papers CORE 2016038, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Schulze, Tim & Grothey, Andreas & McKinnon, Ken, 2017. "A stabilised scenario decomposition algorithm applied to stochastic unit commitment problems," European Journal of Operational Research, Elsevier, vol. 261(1), pages 247-259.
    7. Sven Leyffer & Charlie Vanaret, 2020. "An augmented Lagrangian filter method," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(2), pages 343-376, October.

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