Quasi-Monte Carlo methods for linear two-stage stochastic programming problems

Abstract

Quasi-Monte Carlo algorithms are studied for generating scenarios to solve two-stage linear stochastic programming problems. Their integrands are piecewise linear-quadratic, but do not belong to the function spaces consideredfor 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 second order mixed derivativesexist almost everywhere and belong to $L_2$. This implies that randomly shifted latticerules may achieve the optimal rate of convergence $O(n^{-1+\delta})$ with $\delta \in (0,\frac{1}{2}]$ and a constant not depending on the dimension if the effective superposition dimension is less than or equal to two. The geometric condition is shown to be satisfied for almost all covariance matrices if the underlying probability distribution isnormal. We discuss effective dimensions and techniques for dimension reduction.Numerical experiments for a production planning model with normal inputs showthat indeed convergence rates close to the optimal rate are achieved when usingrandomly shifted lattice rules or scrambled Sobol' point sets accompanied withprincipal component analysis for dimension reduction.