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

Miles Lubin (mlubin***at***mcs.anl.gov)
J. A. Julian Hall (j.a.j.hall***at***ed.ac.uk)
Cosmin G. Petra (petra***at***mcs.anl.gov)
Mihai Anitescu (anitescu***at***mcs.anl.gov)

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 method, parallel computing, stochastic optimization, block-angular

Category 1: Linear, Cone and Semidefinite Programming (Linear Programming )

Category 2: Optimization Software and Modeling Systems (Parallel Algorithms )

Category 3: Stochastic Programming

Citation: Computational Optimization and Applications, in press

Download: [PDF]

Entry Submitted: 04/17/2012
Entry Accepted: 04/17/2012
Entry Last Modified: 02/09/2013

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