A global optimization method based on fuzzy clustering Kriging for aerobic wastewater treatment

Kriging-assisted sequential optimization effectively alleviates the computational burden of complex engineering design problems. Nevertheless, obtaining a globally approximate optimal solution that meets accuracy requirements while minimizing the number of expensive black-box evaluations remains a bottleneck. To address this issue, a global optimization method based on Fuzzy Clustering Kriging (GO-FCK) is proposed. This method employs Latin Hypercube Design (LHD) and iterative sampling to construct a global Kriging model. Subsequently, the Kriging method is used to perform low-cost evaluations on randomly sampled Monte Carlo points, with the points predicted to have the smallest objective values being subjected to fuzzy clustering. The cluster centers and geometric centroid are then determined. Additionally, the point with the minimum Kriging objective value among both the cluster center and the geometric center is regarded as the core of the most promising region. Within this region, a local Kriging model is constructed, and the local optimum is explored by maximizing the Expected Improvement (EI). The aforementioned steps are executed within a loop until the termination condition is met. Testing and comparison on 13 benchmark functions as well as an aerobic wastewater treatment model demonstrate that the proposed method excels in optimization efficiency, robustness, and global convergence.