Computational Optimization and Applications

, Volume 55, Issue 3, pp 571–596

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

  • Miles Lubin
  • J. A. Julian Hall
  • Cosmin G. Petra
  • Mihai Anitescu
Article

DOI: 10.1007/s10589-013-9542-y

Cite this article as:
Lubin, M., Hall, J.A.J., Petra, C.G. et al. Comput Optim Appl (2013) 55: 571. doi:10.1007/s10589-013-9542-y

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.

Keywords

Simplex methodParallel computingStochastic optimizationBlock-angular

Copyright information

© Springer Science+Business Media New York (outside the USA) 2013

Authors and Affiliations

  • Miles Lubin
    • 1
  • J. A. Julian Hall
    • 2
  • Cosmin G. Petra
    • 1
  • Mihai Anitescu
    • 1
  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonneUSA
  2. 2.School of MathematicsUniversity of EdinburghEdinburghUK