In this study, we address the problem of optimizing multi-output black-box functions under uncertain environments. We formulate this problem as the estimation of the uncertain Pareto-frontier (PF) of a multi-output Bayesian surrogate model with two types of variables: design variables and environmental variables. We consider this problem within the context of Bayesian optimization (BO) under uncertain environments, where the design variables are controllable, whereas the environmental variables are assumed to be random and not controllable. The challenge of this problem is to robustly estimate the PF when the distribution of the environmental variables is unknown, that is, to estimate the PF when the environmental variables are generated from the worst possible distribution. We propose a method for solving the BO problem by appropriately incorporating the uncertainties of the environmental variables and their probability distribution. We demonstrate that the proposed method can find an arbitrarily accurate PF with high probability in a finite number of iterations. We also evaluate the performance of the proposed method through numerical experiments.

翻译：本研究针对不确定环境下多输出黑箱函数的优化问题。我们将该问题定义为：在包含设计变量和环境变量两类变量的多输出贝叶斯代理模型中，估计不确定的Pareto前沿（PF）。我们在不确定环境下的贝叶斯优化（BO）框架中考虑该问题，其中设计变量可控，而环境变量视为随机的且不可控。该问题的挑战在于当环境变量分布未知时（即环境变量来自最坏可能分布时）鲁棒地估计PF。我们提出了一种通过恰当融合环境变量及其概率分布不确定性来求解BO问题的方法。我们证明了所提方法能在有限迭代次数内以高概率找到任意精度的PF。我们还通过数值实验评估了所提方法的性能。