两阶段随机规划的拟蒙特卡罗方法有效吗？
Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?
摘要：
研究了两阶段线性随机规划离散逼近设计的拟蒙特卡罗方法。它们的被积函数是分段线性的，但既不光滑，也不位于QMC误差分析所考虑的函数空间中。证明了在弱几何条件下，两阶段模型的方差分解除最高阶项外，其余项都是光滑的。因此，拟蒙特卡罗方法可以利用$δ\in(0,\frac{1}{2}]$和不依赖于维数的常数来实现$O(n^{-1+δ})$的最佳收敛速率。几何条件被证明是一般满足的，如果基本分布是正常的。讨论了两阶段被积函数的灵敏度指标、有效维数和降维技术。数值实验表明，当使用随机置乱的Sobol点集和随机移位的格点规则并辅以适当的降维技术时，确实可以达到接近最优速度的收敛速度。
英文摘要：
Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs. Their 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 smooth. Hence, Quasi-Monte Carlo algorithms may achieve the optimal rate of convergence $O(n^{-1+δ})$ with $δ\in(0,\frac{1}{2}]$ and a constant not depending on the dimension. The geometric condition is shown to be generically satisfied if the underlying distribution is normal. We discuss sensitivity indices, effective dimensions and dimension reduction techniques for two-stage integrands. Numerical experiments show that indeed convergence rates close to the optimal rate are achieved when using randomly scrambled Sobol' point sets and randomly shifted lattice rules accompanied with suitable dimension reduction techniques.
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