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

• Published in: Computational optimization and applications Vol. 55; no. 3; pp. 571 - 596
• Main Authors: Lubin, Miles, Hall, J. A. Julian, Petra, Cosmin G., Anitescu, Mihai
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.07.2013; Springer Nature B.V
• Subjects: 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
• DOI: 10.1007/s10589-013-9542-y

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.