Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

Abstract

Quasi-Monte Carlo algorithms are studied for designing discrete approximationsof two-stage linear stochastic programs. Their integrands are piecewiselinear, but neither smooth nor lie in the function spaces considered for QMC erroranalysis. We show that under some weak geometric condition on the two-stagemodel all terms of their ANOVA decomposition, except the one of highest order,are smooth. Hence, Quasi-Monte Carlo algorithms may achieve the optimal rateof convergence $O(n^{-1+\delta}$ with $\delta \in (0,\frac{1}{2}]$ and a constant not depending on the dimension. The geometric condition is shown to be generically satisfied if the underlyingdistribution is normal. We discuss sensitivity indices, effective dimensionsand dimension reduction techniques for two-stage integrands. Numerical experimentsshow that indeed convergence rates close to the optimal rate are achievedwhen using randomly scrambled Sobol' point sets and randomly shifted latticerules accompanied with suitable dimension reduction techniques.