Constrained Bayesian optimization optimizes a black-box objective function subject to black-box constraints. For simplicity, most existing works assume that multiple constraints are independent. To ask, when and how does dependence between constraints help?, we remove this assumption and implement probability of feasibility with dependence (Dep-PoF) by applying multiple output Gaussian processes (MOGPs) as surrogate models and using expectation propagation to approximate the probabilities. We compare Dep-PoF and the independent version PoF. We propose two new acquisition functions incorporating Dep-PoF and test them on synthetic and practical benchmarks. Our results are largely negative: incorporating dependence between the constraints does not help much. Empirically, incorporating dependence between constraints may be useful if: (i) the solution is on the boundary of the feasible region(s) or (ii) the feasible set is very small. When these conditions are satisfied, the predictive covariance matrix from the MOGP may be poorly approximated by a diagonal matrix and the off-diagonal matrix elements may become important. Dep-PoF may apply to settings where (i) the constraints and their dependence are totally unknown and (ii) experiments are so expensive that any slightly better Bayesian optimization procedure is preferred. But, in most cases, Dep-PoF is indistinguishable from PoF.

约束贝叶斯优化优化受黑盒约束的黑盒目标函数。为简单起见，大多数现有工作假设多个约束是独立的。要问，约束之间的依赖何时以及如何发挥作用？，我们删除了这个假设，并通过应用多个输出高斯过程（MOGP）作为代理模型并使用期望传播来近似概率来实现具有依赖性的可行性概率（Dep-PoF）。我们比较Dep-PoF和独立版本PoF。我们提出了两个结合 Dep-PoF 的新采集函数，并在综合和实际基准上测试它们。我们的结果很大程度上是负面的：合并约束之间的依赖性并没有多大帮助。根据经验，如果出现以下情况，合并约束之间的依赖性可能会很有用：(i) 解位于可行区域的边界上或 (ii) 可行集非常小。当这些条件满足时，MOGP 的预测协方差矩阵可能很难用对角矩阵近似，并且非对角矩阵元素可能变得很重要。Dep-PoF 可能适用于以下情况的设置：(i) 约束及其依赖性完全未知，并且 (ii) 实验成本非常昂贵，因此任何稍微更好的贝叶斯优化程序都是首选。但是，在大多数情况下，Dep-PoF 与 ​​PoF 没有区别。