Abstract
Weight optimization of frame structures with continuous cross-section parametrization is a challenging non-convex problem that has traditionally been solved by local optimization techniques. Here, we exploit its inherent semi-algebraic structure and adopt the Lasserre hierarchy of relaxations to compute the global minimizers. While this hierarchy generates a natural sequence of lower bounds, we show, under mild assumptions, how to project the relaxed solutions onto the feasible set of the original problem and thus construct feasible upper bounds. Based on these bounds, we develop a simple sufficient condition of global \(\varepsilon \)-optimality. Finally, we prove that the optimality gap converges to zero in the limit if the set of global minimizers is convex. We demonstrate these results by means of two academic illustrations.
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Notes
Notice, however, that the solution \(\tilde{\textbf{u}}_j\) to the transformed system \(\begin{pmatrix} \textbf{U}_{\textrm{R},j}^\textrm{T}&\textbf{U}_{\textrm{N},j}^\textrm{T} \end{pmatrix}^\textrm{T} \textbf{K}_j(\textbf{a}) \begin{pmatrix} \textbf{U}_{\textrm{R},j}&\textbf{U}_{\textrm{N},j} \end{pmatrix} \tilde{\textbf{u}}_j = \begin{pmatrix} \textbf{U}_{\textrm{R},j}^\textrm{T}&\textbf{U}_{\textrm{N},j}^\textrm{T} \end{pmatrix}^\textrm{T}\textbf{f}_j\) differs from \(\textbf{u}_j\) in (4b). The original vector field \(\textbf{u}_j\) can be recovered by another transformation as \(\textbf{u}_j = \begin{pmatrix} \textbf{U}_{\textrm{R},j}&\textbf{U}_{\textrm{N},j} \end{pmatrix}\tilde{\textbf{u}}_j\).
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Acknowledgements
We thank the anonymous reviewers for helping us to improve the manuscript presentation.
Funding
Marek Tyburec and Michal Kočvara acknowledge the support of the Czech Science foundation through project No. 22-15524 S. Martin Kružík appreciated the support of the Czech Science Foundation via project No. 21-06569K, and by the Ministry of Education, Youth and Sports through the mobility project 8J20FR019. We also acknowledge support by European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Actions, grant agreement 813211 (POEMA).
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Source codes are available at (Tyburec et al. 2022).
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Tyburec, M., Kočvara, M. & Kružík, M. Global weight optimization of frame structures with polynomial programming. Struct Multidisc Optim 66, 257 (2023). https://doi.org/10.1007/s00158-023-03715-5
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DOI: https://doi.org/10.1007/s00158-023-03715-5