Metamodel-Based Global Optimization Methodologies for High Dimensional Expensive Black-box Problems

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High Dimension
Sensitivity Analysis

Many engineering problems involve high dimensional, computationally expensive, and black-box (HEB) functions such as complex finite element analyses or computational fluid dynamics simulations. Global optimization on HEB problems is challenging since it generally requires a large number of computationally expensive simulations. The aim of this thesis is to tackle optimization on HEB problems. Metamodels are mathematical models that are constructed to approximate black-box and expensive functions. Survey of existing techniques shows that metamodel-based optimization has potential to solve HEB optimization problems. High Dimensional Model Representation (HDMR) metamodeling is chosen from the literature, which is subsequently developed to Principal Component Analysis based HDMR in support of random non-uniform sampling. Also, an adaptively changing basis functions strategy is defined to ensure the orthogonality of basis functions with respect to existing samples in order to achieve the best approximation accuracy for Random Sampling HDMR. Variable correlations revealed through metamodeling are used for decomposing high dimensional problems into smaller sub-problems. Then, sensitivity analysis is used to quantify the intensities of the correlations. For problems in which all correlations are weak or there are a mix of weak and strong correlations, the proposed method is very effective in reducing the total number of function evaluations to achieve a similar accuracy. For problems whose correlations are all strong, the proposed method is not be advantageous. An optimization strategy based on iterative metamodel-supported decomposition is proposed in which the decomposition and optimization phases are performed simultaneously and iteratively, in contrast to a one-time decomposition-optimization process. The results show that except for the category of non-decomposable problems with all or lots of strong correlations, the proposed strategy improves the accuracy of the optimization results noticeably. The developed algorithm is applied to a practical engineering problem to test its effectiveness in real-world applications. An optimal assembly planning problem with a 100-dimensional objective function is optimized using the proposed method. Comparison with other optimization methods results and the baseline values shows significant improvement in the obtained optimum with the same number of function calls. The results represent the state-of-the-art for optimization of HEB problems.

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G. Gary Wang
Applied Sciences:
Thesis type: 
(Thesis) Ph.D.