Optimization under uncertainty is important in many applications, particularly to inform policy and decision making in areas such as public health. A key source of uncertainty arises from the incorporation of environmental variables as inputs into computational models or simulators. Such variables represent uncontrollable features of the optimization problem and reliable decision making must account for the uncertainty they propagate to the simulator outputs. Often, multiple, competing objectives are defined from these outputs such that the final optimal decision is a compromise between different goals. Here, we present emulation-based optimization methodology for such problems that extends expected quantile improvement (EQI) to address multi-objective optimization. Focusing on the practically important case of two objectives, we use a sequential design strategy to identify the Pareto front of optimal solutions. Uncertainty from the environmental variables is integrated out using Monte Carlo samples from the simulator. Interrogation of the expected output from the simulator is facilitated by use of (Gaussian process) emulators. The methodology is demonstrated on an optimization problem from public health involving the dispersion of anthrax spores across a spatial terrain. Environmental variables include meteorological features that impact the dispersion, and the methodology identifies the Pareto front even when there is considerable input uncertainty.

翻译：不确定性条件下的优化在许多应用中具有重要意义，尤其在公共卫生等领域的政策制定与决策支持中。环境变量作为计算模型或模拟器的输入参数是产生不确定性的关键来源。这类变量构成了优化问题中不可控的特征，可靠的决策必须考虑这些变量对模拟器输出传播的不确定性。通常，研究者会根据模拟器输出定义多个相互竞争的目标函数，使得最终最优决策成为不同目标之间的折衷方案。本文针对此类问题提出了一种基于代理模型的优化方法，将期望分位数改进扩展到多目标优化场景。聚焦于两个目标这一具有重要实践意义的情形，我们采用序贯设计策略识别帕累托最优解前沿。通过模拟器的蒙特卡洛采样积分，消解环境变量引入的不确定性。借助高斯过程代理模型实现模拟器期望输出的高效查询。该方法的有效性通过公共卫生领域的优化问题得到验证——该问题涉及炭疽芽孢在地形空间中的扩散传播。环境变量涵盖影响扩散过程的气象特征，即使在输入存在显著不确定性的条件下，该方法仍能成功识别帕累托前沿。