Invited Session Fri.3.MA 141

Friday, 15:15 - 16:45 h, Room: MA 141

Cluster 22: Stochastic optimization [...]

Scenario generation in stochastic programming


Chair: Mihai Anitescu



Friday, 15:15 - 15:40 h, Room: MA 141, Talk 1

Sanjay Mehrotra
New results in scenario generation for stochastic optimization problems via the sparse grid method

Coauthors: Michael Chen, David Papp


We study the use of sparse grid methods for the scenario generation (or discretization) problem in stochastic optimization problems when 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 converges to the true optimal solution as the number of scenarios increases. The rate of convergence is also established. An improvement is presented for stochastic programs in the case when the uncertainty is described using a linear transformation of a product of univariate distributions, such as joint normal distributions. We numerically compare the performance of sparse grid methods with quasi-Monte Carlo and Monte Carlo scenario generation. The results show that the sparse grid method is very efficient if the integrand is sufficiently smooth, and that the method is potentially scalable to thousands of random variables.



Friday, 16:15 - 16:40 h, Room: MA 141, Talk 3

Tito Homem-De-Mello
On scenario generation methods for a hydroelectric power system

Coauthors: Vitor L. de Matos, Erlon C. Finardi


We study a multi-stage stochastic programming model for hydrothermal energy planning in Brazil, where uncertainty is due to water inflows. We discuss some methods for generation of scenario trees that can be used by an optimization algorithm to solve the problem. Although the original input process is modeled with an periodic auto-regressive model, by making a transformation we can reduce the input process to one that is stage-wise independent. That in turn allows us to proceed with generating scenarios stage-by-stage following the
approach in Mirkov and Pflug (2007). The algorithm we propose hinges on the stage-wise independence property and consists of two phases: first, we generate a scenario tree where the distribution in each stage is approximated by a discrete distribution with large number of points; then, we apply a reduction method to find a distribution with smaller support that minimizes the Wasserstein distance to that discrete distribution. We show how this minimization problem can be solved with a structured binary linear program. Some numerical results are presented to illustrate the ideas.


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