This article considers nonconvex global optimization problems subject to uncertainties described by continuous random variables. Such problems arise in chemical process design, renewable energy systems, stochastic model predictive control, and many other applications. Here, we restrict our attention to problems with expected‐value objectives and no recourse decisions. In principle, such problems can be solved globally using spatial branch‐and‐bound. However, branch‐and‐bound requires the ability to bound the optimal objective value on subintervals of the search space, and existing techniques are not generally applicable because expected‐value objectives often cannot be written in closed‐form. To address this, this article presents a new method for computing convex and concave relaxations of nonconvex expected‐value functions, which can be used to obtain rigorous bounds for use in branch‐and‐bound. Furthermore, these relaxations obey a second‐order pointwise convergence property, which is sufficient for finite termination of branch‐and‐bound under standard assumptions. Empirical results are shown for three simple examples. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3023–3033, 2018

中文翻译：

---

连续随机变量描述的不确定性下全局优化的凸松弛

本文考虑受连续随机变量描述的不确定性影响的非凸全局优化问题。这样的问题出现在化学过程设计，可再生能源系统，随机模型预测控制以及许多其他应用中。在这里，我们将注意力集中在具有预期价值目标且无追索权决策的问题上。原则上，可以使用空间分支定界法全局解决此类问题。但是，分支定界要求有能力将最佳目标值限制在搜索空间的子间隔上，并且现有技术通常不适用，因为期望值目标通常无法以封闭形式编写。为了解决这个问题，本文提出了一种计算非凸期望值函数的凸和凹弛豫的新方法，可用于获取严格的界限，以用于分支定界。此外，这些松弛服从二阶逐点收敛性，这足以满足标准假设下分支定界的有限终止。给出了三个简单示例的经验结果。©2018美国化学工程师学会AIChE J，64：3023–3033，2018