Abstract
Design optimization of high-dimensional, computationally-expensive, and black-box problems has many challenges to resolve. Owing to the enormously high computational costs, obtaining the optimum design that has the minimum or maximum objective function value while satisfying constraints requires elaborated approaches. Instead, this paper suggests that if a design that satisfies a design target demanded by designers or decision makers can be obtained with low computational costs, the design could be quite useful for practitioners. To minimize the computational costs while obtaining the design, we aim to select the minimum number of significant input variables of a high-dimensional problem. Accordingly, a new design optimization problem for input variable selection, which is named design-target-based optimization (DTBO), is proposed to achieve it. The input variable selection is performed based on the selection measure that is calculated by the significance of input variables and the weights of each response. In the viewpoint of obtaining the design by minimally selecting the significant input variables, it is crucial to allocate weights appropriately by considering the status of constraints whether they are violated, active or inactive. Therefore, a penalty-Lagrange multiplier (PLM) method is also proposed for the DTBO to allocate weights adaptively. The performance and effectiveness of the DTBO for solving high-dimensional design problems are demonstrated by examining two numerical examples and the design of the body-in-white of a vehicle. We anticipate that the DTBO with the PLM method can obtain a practical design satisfying the design target efficiently.
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Acknowledgements
The authors would like to acknowledge the financial support from the BK21 Four Education and Research Program for Automotive-Software Convergence funded by the Korean Government.
Funding
This work was supported by the BK21 Four Education and Research Program for Automotive-Software Convergence funded by the Korean Government (Grant No. 5199990814043).
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All authors contributed to the study conception and design and performed material preparation, data collection, and analysis. The first draft of the manuscript was written by HK and THL provided supervision, revised, edited, and commented on all the versions of the manuscript. All authors read and approved the final manuscript.
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Replication of results
For readers interested in the specific process of the DTBO using input variable selection, Sects. 2 and 3 explain the basic principles. Kim and Lee’s paper also provides detailed explanations of the RENBOOT for significance analysis (Kim and Lee 2021), which was implemented using R statistical software. Algorithm 1 shows the main workflow of the code implementation for the DTBO using input variable selection. In particular, the selection measure and weight allocation can be calculated using Eqs. (11) to (15). The authors used lhsdesign with the maximin criterion (OLHD) in MATLAB 2021b for the initial dataset. For the kriging surrogate model, the DACE MATLAB toolbox (Lophaven et al. 2002) was used, and the MDD (Johnson et al. 1990) was used for the additional data points. Sequential quadratic programming in MATLAB 2021b was then used for optimization in the DTBO. The FE model of the BIW of a vehicle can be downloaded from LSTC (Livermore Software Technology Corporation 2011).
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Kim, H., Lee, T.H. Design-target-based optimization using input variable selection and penalty-Lagrange multiplier for high-dimensional design problems. Struct Multidisc Optim 65, 258 (2022). https://doi.org/10.1007/s00158-022-03356-0
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DOI: https://doi.org/10.1007/s00158-022-03356-0