Fuzzy clustering based hierarchical metamodeling for design optimization

Metamodeling techniques are becoming commonly used for reducing the computational burden of many design optimization problems; 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 paper proposes an intuitive methodology to systematically reduce the design space to a relatively small region of interest. The methodology entails three main elements: 1) applying the response surface methodology to capture the behavior of unknown functions in the original large space; 2) calculating many inexpensive points from the obtained response surface, clustering these points using the Fuzzy c-means clustering method, and choosing the attractive cluster and its corresponding design space; 3) in the reduced space, progressively generating sampling points to build kriging models and to identify the design optimum. The proposed methodology is illustrated using the wellknown six-hump camel back problem; a constrained, highly-nonlinear, 1-D optimization problem; and a real engineering design problem. It is found that the proposed methodology can intuitively capture promising design regions and can efficiently identify the global or near-global design optimum even in the presence of highly nonlinear constraints. Potential difficulties and further research topics are also discussed.