We consider the problem of chance constrained optimization where it is sought to optimize a function and satisfy constraints, both of which are affected by uncertainties. The real world declinations of this problem are particularly challenging because of their inherent computational cost. To tackle such problems, we propose a new Bayesian optimization method. It applies to the situation where the uncertainty comes from some of the inputs, so that it becomes possible to define an acquisition criterion in the joint controlled-uncontrolled input space. The main contribution of this work is an acquisition criterion that accounts for both the average improvement in objective function and the constraint reliability. The criterion is derived following the Stepwise Uncertainty Reduction logic and its maximization provides both optimal controlled and uncontrolled parameters. Analytical expressions are given to efficiently calculate the criterion. Numerical studies on test functions are presented. It is found through experimental comparisons with alternative sampling criteria that the adequation between the sampling criterion and the problem contributes to the efficiency of the overall optimization. As a side result, an expression for the variance of the improvement is given.

翻译：考虑机会约束优化问题，即在满足约束条件的同时优化目标函数，而两者均受不确定性影响。该问题的实际应用场景因其固有的计算成本而极具挑战性。为解决此类问题，我们提出一种新型贝叶斯优化方法。该方法适用于不确定性来自部分输入变量的情形，从而能够在联合受控-非受控输入空间中定义采集准则。本研究的主要贡献在于提出一种同时考虑目标函数平均改进量与约束可靠性的采集准则。该准则基于逐步不确定性削减逻辑推导得出，其最大化过程可同步获取最优受控参数与非受控参数。本文给出了该准则的高效解析表达式，并通过测试函数数值实验展开研究。与替代采样准则的实验对比表明：采样准则与问题间的匹配程度直接影响整体优化效率。作为附加成果，本文给出了改进量方差的解析表达式。