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

Abstract

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives and belong to $$L_{2}$$L2. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence $$O(n^{-1+\delta })$$O(n-1+ź) with $$\delta \in (0,\frac{1}{2}]$$źź(0,12] and a constant not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction.