Parallel distributedmemory simplex for largescale stochastic LP problems
 Miles Lubin,
 J. A. Julian Hall,
 Cosmin G. Petra,
 Mihai Anitescu
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We present a parallelization of the revised simplex method for large extensive forms of twostage 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, interiorpoint methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hotstarts (e.g., in a branchandbound context) and can provide important sensitivity information. Our approach exploits the dual blockangular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for highperformance distributedmemory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual blockangular structure. Our implementation is competitive in serial with highly efficient sparsityexploiting simplex codes and achieves significant relative speedups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largestscale parallel sparsityexploiting 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 blockangular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying productform updates.
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 Title
 Parallel distributedmemory simplex for largescale stochastic LP problems
 Journal

Computational Optimization and Applications
Volume 55, Issue 3 , pp 571596
 Cover Date
 20130701
 DOI
 10.1007/s105890139542y
 Print ISSN
 09266003
 Online ISSN
 15732894
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Simplex method
 Parallel computing
 Stochastic optimization
 Blockangular
 Industry Sectors
 Authors

 Miles Lubin ^{(1)}
 J. A. Julian Hall ^{(2)}
 Cosmin G. Petra ^{(1)}
 Mihai Anitescu ^{(1)}
 Author Affiliations

 1. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 604394844, USA
 2. School of Mathematics, University of Edinburgh, JCMB, King’s Buildings, Edinburgh, EH9 3JZ, UK