SIAM Journal on Optimization, Volume 30, Issue 2, Page 1119-1143, January 2020.
We treat the so-called scenario approach, a popular probabilistic approximation method for robust minmax optimization problems via independent and identically distributed (i.i.d.) sampling from the uncertainty set, from various perspectives. The scenario approach is well studied in the important case of convex robust optimization problems, and here we examine how the phenomenon of concentration of measures affects the i.i.d. sampling aspect of the scenario approach in high dimensions and its relation with the optimal values. Moreover, we perform a detailed study of both the asymptotic behavior (consistency) and finite time behavior of the scenario approach in the more general setting of nonconvex minmax optimization problems. In the direction of the asymptotic behavior of the scenario approach, we present an obstruction to consistency that arises when the decision set is noncompact. In the direction of finite sample guarantees, we establish a general methodology for extracting “probably approximately correct''-type estimates for the finite sample behavior of the scenario approach for a large class of nonconvex problems.

中文翻译：

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以非凸案例为重点的Minmax优化的场景方法：正面结果和警告

SIAM优化杂志，第30卷，第2期，第1119-1143页，2020年1月。
我们处理所谓的场景方法，这是一种流行的概率逼近方法，用于通过各种角度从不确定性集中进行独立且均匀分布的（iid）采样来解决鲁棒的minmax优化问题。在凸鲁棒优化问题的重要情况下，对方案方法进行了很好的研究，在这里，我们研究了度量集中现象如何在高维度上影响方案方法的iid采样方面及其与最佳值的关系。此外，在非凸minmax优化问题的更一般设置中，我们对方案方法的渐近行为（一致性）和有限时间行为进行了详细研究。在场景方法的渐近行为的方向上，我们提出了当决策集不紧凑时会出现一致性障碍。在有限样本保证的方向上，我们建立了一种通用方法，用于针对大类非凸问题的情景方法的有限样本行为提取“可能近似正确”的类型估计。