Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and
continuous probability distribution. The latter should allow for a transformation to a
distribution with independent marginals. The two-stage integrands are piecewise linear,
but neither smooth nor lie in 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 that first and second order ANOVA terms have mixed first order partial
derivatives and belong to L2 . Hence, randomly shifted lattice rules (SLR) may achieve
the optimal rate of convergence O(n−1+δ ) with δ ∈ (0, 12 ] and a constant not depending
on the dimension if the effective superposition dimension is at most two. We discuss
effective dimensions and dimension reduction for two-stage integrands. The geometric
condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming
the covariance matrix. Numerical experiments for a large scale two-stage stochastic
production planning model with normal demand show that indeed convergence rates
close to the optimal are achieved when using SLR and randomly scrambled Sobol’ point
sets accompanied with PCA for dimension reduction.
Natural gas is important for the energy turnaround in many countries like in Germany, where it serves as a "bridging energy" towards a fossil-free energy supply in the future. About 20% of the total German energy demand is provided by natural gas, which is transported through a complex pipeline network with a total length of about 30000 km and the efficient use of the given transport infrastructure for natural gas is of political, economic, and societal importance.
As a consequence of the liberalization of the European gas market in the last decades, gas trading and transport have been decoupled. This has led to new challenges for gas transport companies, and mathematical optimization is perfectly suited for tackling many of these challenges. However, the underlying mathematical problems are by far too hard to be solved by today's general-purpose software so that novel mathematical theory and algorithms are needed. The industrial research project "ForNe: Research Cooperation Network Optimization" has been initiated and funded by Open Grid Europe in 2009 and brought together experts in mathematical optimization from seven German universities and research institutes, which cover almost the entire range of mathematical optimization: integer and nonlinear optimization as well as optimization under uncertainty.
The mathematical research results have been put together in a software package that has been delivered to Open Grid Europe at the end of the project. Moreover, the research is still continuing - e.g., in the Collaborative Research Center/Transregio 154 "Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks" funded by the German Research Foundation.