This paper deals with the scenario approach to robust optimization. This relies on a random sampling of the possibly infinite number of constraints induced by uncertainties in the parameters of an optimization problem. Solving the resulting random program yields a solution for which the quality is measured in terms of the probability of violating the constraints for a random value of the uncertainties, typically unseen before. Another central issue is the determination of the sample complexity, i.e., the number of random constraints (or scenarios) that one must consider in order to guarantee a certain level of reliability. In this paper, we introduce the notion of margin to improve upon standard results in this field. In particular, using tools from statistical learning theory, we show that the sample complexity of a class of random programs does not explicitly depend on the number of variables. In addition, within the considered class, that includes polynomial constraints among others, this result holds for both convex and nonconvex instances with the same level of guarantees. We also derive a posteriori bounds on the probability of violation and sketch a regularization approach that could be used to improve the reliability of computed solutions on the basis of these bounds.

翻译：本文研究鲁棒优化的场景方法。该方法通过对优化问题中参数不确定性所引发的可能无限个约束进行随机采样。求解所得随机规划得到的解，其质量通过随机不确定性值（通常未见过的）违反约束的概率来衡量。另一个核心问题是确定样本复杂度，即为保证一定可靠性水平所需考虑的随机约束（或场景）数量。本文引入边际概念以改进该领域的标准结果。特别是，利用统计学习理论工具，我们证明一类随机规划的样本复杂度并不显式依赖于变量数量。此外，在所考虑的类别（包括多项式约束等）中，该结论对凸和非凸实例均成立，且保证相同水平的可靠性。我们还推导了违反概率的后验界，并基于这些界勾勒了一种可用于提高计算解可靠性的正则化方法框架。