Uncertainty is prevalent in engineering design, data-driven problems, and decision making broadly. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving conservative optimization models expressed using measures of risk and related concepts. We survey the rapid development of risk measures over the last quarter century. From their beginning in financial engineering, we recount the spread to nearly all areas of engineering and applied mathematics. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. We describe the key facts, list several concrete algorithms, and provide an extensive list of references for further reading. The survey recalls connections with utility theory and distributionally robust optimization, points to emerging applications areas such as fair machine learning, and defines measures of reliability.

翻译：不确定性广泛存在于工程设计、数据驱动问题以及广义决策过程中。由于固有的风险规避倾向及对模型假设的模糊性，常见应对策略是通过构建并求解以风险度量及其相关概念表达的保守优化模型。本文系统综述了过去二十五年间风险度量的快速发展历程。从其发源于金融工程领域，我们追溯了该概念向工程学与应用数学几乎所有领域的传播脉络。风险度量植根于凸分析理论，为处理不确定性提供了具有显著计算与理论优势的通用框架。我们阐释了关键性质，列举了若干具体算法，并提供了详尽的参考文献列表以供深入研读。本综述梳理了风险度量与效用理论及分布式鲁棒优化的内在联系，指出其在公平机器学习等新兴应用领域的前景，并定义了可靠性度量指标。