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

TL;DR

This paper investigates the efficiency of Quasi-Monte Carlo algorithms for two-stage stochastic programs with non-smooth integrands, demonstrating conditions under which optimal convergence rates are achievable.

Contribution

It establishes that under a weak geometric condition, QMC methods can attain near-optimal convergence rates for two-stage stochastic programs with non-smooth integrands.

Findings

QMC algorithms can achieve near-optimal convergence rates under certain geometric conditions.

Numerical experiments confirm the effectiveness of dimension reduction techniques with QMC.

The geometric condition is generically satisfied for normal distributions.

Abstract

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs. Their 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 smooth. Hence, Quasi-Monte Carlo algorithms 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. The geometric condition is shown to be generically satisfied if the underlying distribution is normal. We discuss sensitivity indices, effective dimensions and dimension reduction techniques for two-stage integrands. Numerical experiments show that indeed convergence rates close to the optimal rate are achieved when using randomly scrambled Sobol' point sets and randomly shifted lattice rules accompanied with suitable dimension reduction techniques.