We consider randomized QMC methods for approximating the expected recourse in two-stage stochastic optimization problems containing mixed-integer decisions in the second stage. It is known that the second-stage optimal value function is piecewise linear-quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets. This structure is exploited to provide conditions implying that first and higher order terms of the integrand’s ANOVA decomposition (Math. Comp. 79 (2010), 953–966) have mixed weak first order partial derivatives. This leads to a good smooth approximation of the integrand and, hence, to good convergence rates of randomized QMC methods if the effective (superposition) dimension is low.

中文翻译：

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两阶段随机混合整数程序的拟蒙特卡罗方法

我们考虑使用随机 QMC 方法来逼近包含第二阶段混合整数决策的两阶段随机优化问题中的预期资源。已知第二阶段最优值函数是分段线性二次函数，在某些凸多面体集的边界处可能存在扭结和不连续性。该结构被用来提供条件，暗示被积函数的 ANOVA 分解 (Math. Comp. 79 (2010), 953–966) 的一阶和高阶项具有混合的弱一阶偏导数。这导致被积函数的良好平滑近似，因此，如果有效（叠加）维数较低，则随机 QMC 方法具有良好的收敛速度。