Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

详细信息    查看全文

• 作者：H. Heitsch ; H. Leövey ; W. Römisch
• 关键词：Stochastic programming ; Two ; stage ; Scenario ; Quasi ; Monte Carlo ; Effective dimension ; Dimension reduction
• 刊名：Computational Optimization and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：65
• 期：3
• 页码：567-603
• 全文大小：797 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Optimization
  Operations Research and Mathematical Programming
  Operation Research and Decision Theory
  Statistics
  Convex and Discrete Geometry
  
• 出版者：Springer Netherlands
• ISSN：1573-2894
• 卷排序：65

文摘

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 \(L_{2}\). Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence \(O(n^{-1+\delta })\) with \(\delta \in (0,\frac{1}{2}]\) 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.