Although parallel efficiency using the revised simplex method has not been achieved for general large sparse LP problems, this talk will show how the particular structure of stochastic LP problems gives scope for efficient data parallelism.
Very large stochastic LP problems have been considered to be too big to solve with the simplex method, with decomposition approaches or, more recently, interior-point methods being preferred. However, these approaches do not provide optimal basic solutions which allow for the efficient hot-starts required, for example, in a branch-and-bound context when solving stochastic MIP problems.
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 the focus is 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 parallel efficiency when using up to 128 cores and runs up to 100 times faster than the leading open-source serial solver. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality: possibly the largest LPs ever solved using the simplex method.
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.
Parallel distributed-memory simplex for large-scale stochastic LP problems
M. Lubin, J. A. J. Hall, C. G. Petra and M. Anitescu
Computational Optimization and Applications 55(3), 571-596, 2013. DOI: 10.1007/s10589-013-9542-y
Parallelizing the dual revised simplex method
Q. Huangfu and J. A. J. Hall
Technical Report ERGO-14-011.