Expected improvement (EI), a function of prediction uncertainty \(\sigma (\mathbf{x})\)and improvement quantity \( {(\xi - {{\hat{y}}}({\mathbf{x}}))}\), has been widely used to guide the Bayesian optimization (BO). However, the EI-based BO can get stuck in sub-optimal solutions even with a large number of samples. The previous studies attribute such sub-optimal convergence problem to the “over-exploitation” of EI. Differently, we argue that, in addition to the “over-exploitation”, EI can also get trapped in querying samples with maximum \( \sigma ({\mathbf{x}})\) but poor objective function value \(y(\mathbf{x})\). We call such issue as “over-exploration”, which can be a more challenging problem that leads to the sub-optimal convergence rate of BO. To address the issues of “over-exploration” and “over-exploitation” simultaneously, we propose to calibrate the incumbent \(\xi \) adaptively instead of fixing it as the present best solution in the EI formulation. Furthermore, we propose two calibrated versions of EI, namely calibrated EI (CEI) and recalibrated EI (REI), which combine the calibrated incumbent \(\xi ^\text{Calibrated}\) with distance constraint to enhance the local exploitation and global exploration of promising areas, respectively. After that, we integrate EI with CEI & REI to devise a novel BO algorithm named as CR-EI. Through tests on seven benchmark functions and an engineering problem of airfoil optimization, the effectiveness of CR-EI has been well demonstrated.

预期改进（EI），预测不确定性的函数\(\sigma (\mathbf{x})\)和改进量\( {(\xi - {{\hat{y}}}({\mathbf{x} }))}\)，已被广泛用于指导贝叶斯优化（BO）。然而，即使有大量样本，基于 EI 的 BO 也可能陷入次优解决方案。先前的研究将这种次优收敛问题归因于EI的“过度开发”。不同的是，我们认为，除了“过度开发”之外，EI 还可能陷入查询最大\( \sigma ({\mathbf{x}})\)但目标函数值较差\(y( \mathbf{x})\). 我们将此类问题称为“过度探索”，这可能是一个更具挑战性的问题，会导致 BO 的收敛速度不理想。为了同时解决“过度探索”和“过度开发”的问题，我们建议自适应地校准现任\(\xi \)，而不是将其固定为当前 EI 公式中的最佳解决方案。此外，我们提出了两个校准版本的 EI，即校准 EI (CEI) 和重新校准 EI (REI)，它们结合了校准的现任者\(\xi ^\text{Calibrated}\) 通过距离约束，分别增强对有前景区域的局部开发和全局探索。之后，我们将 EI 与 CEI 和 REI 相结合，设计出一种名为 CR-EI 的新型 BO 算法。通过对7个基准函数和翼型优化工程问题的测试，CR-EI的有效性得到了很好的证明。