### Werner RĂ¶misch (Humboldt University, Berlin)

#### Quasi-Monte Carlo methods for two-stage stochastic programs

*Wednesday 22 October 2014 at 15.00, JCMB 6206*

##### Abstract

Quasi-Monte Carlo algorithms are studied for generating scenarios to solve
two-stage linear stochastic programs. Their integrands are piecewise
linear-quadratic, but do not belong to 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 continuously differentiable and second order mixed
derivatives exist almost everywhere and are quadratically integrable. This
implies that randomly shifted lattice rules may achieve the optimal rate of
convergence O(n^{-1+δ}) with δ in (0,1/2] and a constant
not depending on the dimension if the effective dimension is equal to two. The
geometric condition is shown to be generically satisfied if the underlying
probability distribution is normal. We discuss effective dimensions and
techniques for dimension reduction. Numerical experiments for a production
planning model with normal inputs show that indeed convergence rates close to
the optimal rate are achieved when using randomly shifted lattice rules or
scrambled Sobol' point sets accompanied with principal component analysis for
dimension reduction.

### Seminars by year

*Current*
*2016*
*2015*
*2014*
*2013*
*2012*
*2011*
*2010*
*2009*
*2008*
*2007*
*2006*
*2005*
*2004*
*2003*
*2002*
*2001*
*2000*
*1999*
*1998*
*1997*
*1996*