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Abstract
While studies in metamodel-assisted optimization predominantly involve continuous variables, this paper explores the additional presence of categorical data, representing for instance the choice of a material or the type of connection. The common approach consisting in mapping them onto integers might lead to inconsistencies or poor approximation results. Therefore, an investigation of the best coding is necessary; however, to build accurate and flexible metamodels, a special attention should also be devoted to the treatment of the distinct nature of the variables involved. Consequently, a multiple kernel regression methodology is proposed, since it allows for selecting separate kernel functions with respect to the variable type. The validation of the advocated approach is carried out on six analytical benchmark test cases and on the structural responses of a rigid frame. In all cases, better performances are obtained by multiple kernel regression with respect to its single kernel counterpart, thereby demonstrating the potential offered by this approach, especially in combination with dummy coding. Finally, multi-objective surrogate-based optimization is performed on the rigid frame example, firstly to illustrate the benefit of dealing with mixed variables for structural design, then to show the reduction in terms of finite element simulations obtained thanks to the metamodels.
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Notes
This paper is based on a contribution presented at the 10th World Congress on Structural and Multidisciplinary Optimization (WCSMO-10), Orlando, Florida, USA, May 19-24, 2013.
For a synthesis of single-objective optimization studies for mixed variables in engineering design, the reader is referred to Filomeno Coelho (2013).
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Acknowledgments
The authors would like to thank the Associate Editor and the Reviewers for their fruitful comments and suggestions.
The second and third authors also acknowledge support by the Basic Project Foundation of Northwestern Polytechnical University (GCKY1011).
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ULB–BATir Department, Université libre de Bruxelles, Avenue F.D. Roosevelt, 50 (CP 194/2), B-1050, Brussels, Belgium
Manuel Herrera & Rajan Filomeno Coelho
NPU–Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, People’s Republic of China
Aurore Guglielmetti & Manyu Xiao
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Correspondence toRajan Filomeno Coelho.
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This work has been supported by Innoviris (Brussels-Capital Region, Belgium) through a BB2B project entitled “Multicriteria optimization with uncertainty quantification applied to the building industry”.
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Herrera, M., Guglielmetti, A., Xiao, M.et al. Metamodel-assisted optimization based on multiple kernel regression for mixed variables.Struct Multidisc Optim49, 979–991 (2014). https://doi.org/10.1007/s00158-013-1029-z
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