随机变分不等式真解的置信区间
Individual confidence intervals for true solutions to stochastic variational inequalities
摘要：
随机变分不等式（SVI）提供了一种建模各种优化和平衡问题的方法，其中数据是不确定的。通常需要通过样本平均近似（SAA）问题的解来估计真实的SVI解。本文提出了三种方法来建立置信区间的组件的真正的解决方案，这些区间是计算从一个单一的SAA解决方案。前两种方法使用“间接方法”，需要初始计算渐近精确的置信区间，用于SVI的法线映射公式的解决方案。第三种方法直接构建真实SVI解决方案的置信区间;在前两种方法适用的相同情况下，该方法产生的区间满足最低指定置信水平。我们证明了这三种方法的理论与弱收敛的结果，讨论如何实现这些方法，并测试其性能使用两个数值例子。
英文摘要：
Stochastic variational inequalities (SVI) provide a means for modeling various optimization and equilibrium problems where data are subject to uncertainty. Often it is necessary to estimate the true SVI solution by the solution of a sample average approximation (SAA) problem. This paper proposes three methods for building confidence intervals for components of the true solution, and those intervals are computable from a single SAA solution. The first two methods use an "indirect approach" that requires initially computing asymptotically exact confidence intervals for the solution to the normal map formulation of the SVI. The third method directly constructs confidence intervals for the true SVI solution; intervals produced with this method meet a minimum specified level of confidence in the same situations for which the first two methods are applicable. We justify the three methods theoretically with weak convergence results, discuss how to implement these methods, and test their performance using two numerical examples.
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