A study of the augmented system and column-splitting approaches for solving two-stage stochastic linear programs by interior-point methods

Linear programs that arise in two-stage stochastic programming offer a particularly difficult test of the robustness of interior-point methods. These LPs are typically very large, yet incorporate ‘dense columns’—corresponding to the first-stage variables—that rule out the standard approach of solving the so-called normal equations. We compare two alternative approaches by running tests on six groups of stochastic linear programs that have been used in previous studies. We find that the augmented system approach consistently requires an amount of work that is very nearly linear in the number of scenarios, whereas the column-splitting approach requires a less predictable amount of work that grows worse than quadratically in some cases. We conclude by explaining how the differences between the two approaches can be viewed as a consequence of the way in which myopic elimination strategies deal with the block-structured matrices involved. Our analysis also leads to a useful comparison of the augmented system approach with the Schur complement approach considered in other recent research.