Scenario generation for stochastic optimization problems via the sparse grid method

We study the use of sparse grids methods for the scenario generation (or discretization) problem in stochastic programming problems where the uncertainty is modeled using a continuous multivariate distribution. We show that, under a regularity assumption on the random function, the sequence of optimal solutions of the sparse grid approximations, as the number of scenarios increases, converges to the true optimal solutions. The rate of convergence is also established. We consider the use of quadrature formulas tailored to the stochastic programs where the uncertainty can be described via a linear transformation of a product of univariate distributions, such as joint normal distributions. We numerically compare the performance of the sparse grid method using different quadrature rules with quasi-Monte Carlo (QMC) methods and Monte Carlo (MC) scenario generation, using a series of utility maximization problems with up to 160 random variables. The results show that the sparse grid method is very efficient if the integrand is sufficiently smooth. In such problems the sparse grid scenario generation method is found to need several orders of magnitude fewer scenarios than MC and QMC scenario generation to achieve the same accuracy. The method is potentially scalable to problem with thousands of random variables.