skip to main content


Title: Scenario generation for stochastic optimization problems via the sparse grid method

We study the use of sparse grids in 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 involved, the sequence of optimal objective function values of the sparse grid approximations converges to the true optimal objective function values as the number of scenarios increases. The rate of convergence is also established. We treat separately the special case when the underlying distribution is an affine transform of a product of univariate distributions, and show how the sparse grid method can be adapted to the distribution by the use of quadrature formulas tailored to the distribution. We numerically compare the performance of the sparse grid method using different quadrature rules with classic quasi-Monte Carlo (QMC) methods, optimal rank-one lattice rules, 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, especially 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 QMCmore » scenario generation to achieve the same accuracy. As a result, it is indicated that the method scales well with the dimension of the distribution--especially when the underlying distribution is an affine transform of a product of univariate distributions, in which case the method appears scalable to thousands of random variables.« less
 [1] ;  [2] ;  [3]
  1. York Univ., Toronto (Canada)
  2. Northwestern Univ., Evanston, IL (United States)
  3. North Carolina State Univ., Raleigh, NC (United States)
Publication Date:
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Computational Optimization and applications
Additional Journal Information:
Journal Volume: 62; Journal Issue: 3; Journal ID: ISSN 0926-6003
Research Org:
Northwestern Univ., Evanston, IL (United States)
Sponsoring Org:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
Country of Publication:
United States
97 MATHEMATICS AND COMPUTING; scenario generation; Stochastic optimization; discretization; sparse grid
OSTI Identifier: