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 space is also compared and contrasted.