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

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 rec...

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Vydáno v:Computational optimization and applications Ročník 55; číslo 3; s. 571 - 596
Hlavní autoři: Lubin, Miles, Hall, J. A. Julian, Petra, Cosmin G., Anitescu, Mihai
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.07.2013
Springer Nature B.V
Témata:
Algorithms
Blocking
Computation
Computer science
Convex and Discrete Geometry
Decomposition
Linear algebra
Linear programming
Management Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Methods
Operations Research
Operations Research/Decision Theory
Optimization
Random variables
Simplex method
Sparsity
Statistics
Stochasticity
Studies
Parallel computing
Stochastic optimization
Simplex method
Block-angular
ISSN:0926-6003, 1573-2894
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Shrnutí: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.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-013-9542-y