Fuzzy clustering based hierarchical metamodeling for design space reduction and optimization

For computation-intensive design problems, metamodeling techniques are commonly used to reduce the computational expense during optimization; however, they often have difficulty or even fail to model an unknown system in a large design space, especially when the number of available samples is limited. This article proposes an intuitive methodology to systematically reduce the design space to a relatively small region. This methodology entails three main elements: (1) constructing metamodels using either response surface or kriging models to capture unknown system behavior in the original large space; (2) calculating many inexpensive points from the obtained metamodel, clustering these points using the fuzzy c-means clustering method, and choosing an attractive cluster and its corresponding reduced design space; (3) progressively generating sample points to construct kriging models and identify the design optimum within the reduced design space. The proposed methodology is illustrated using the well-known six-hump camel back problem, a highly nonlinear constrained optimization problem, and a real design problem. Through comparison with other methods, it is found that the proposed methodology can intuitively capture promising design regions and can efficiently identify the global or near-global design optimum in the presence of highly nonlinear constraints. The effect of using either response surface or kriging models in the original design space is also compared and contrasted. Limitations of the proposed methodology are discussed.