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

Journal Article · · Computational Optimization and Applications
 [1];  [2];  [3]
  1. York Univ., Toronto (Canada)
  2. Northwestern Univ., Evanston, IL (United States)
  3. North Carolina State Univ., Raleigh, NC (United States)
+ Show Author Affiliations

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 QMC 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.

Research Organization:
Northwestern Univ., Evanston, IL (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
Grant/Contract Number:
SC0005102
OSTI ID:
1321132
Journal Information:
Computational Optimization and Applications, Vol. 62, Issue 3; ISSN 0926-6003
Publisher:
SpringerCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 10 works
Citation information provided by
Web of Science

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Fast scenario reduction by conditional scenarios in two-stage stochastic MILP problems journal December 2019

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Related Subjects

97 MATHEMATICS AND COMPUTING
scenario generation
Stochastic optimization
discretization
sparse grid