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Truss optimization with discrete design variables: a critical review

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Abstract

This review presents developed models, theory, and numerical methods for structural optimization of trusses with discrete design variables in the period 1968 – 2014. The comprehensive reference list collects, for the first time, the articles in the field presenting deterministic optimization methods and meta heuristics. The field has experienced a shift in focus from deterministic methods to meta heuristics, i.e. stochastic search methods. Based on the reported numerical results it is however not possible to conclude that this shift has improved the competences to solve application relevant problems. This, and other, observations lead to a set of recommended research tasks and objectives to bring the field forward. The development of a publicly available benchmark library is urgently needed to support development and assessment of existing and new heuristics and methods. Combined with this effort, it is recommended that the field begins to use modern methods such as performance profiles for fair and accurate comparison of optimization methods. Finally, theoretical results are rare in this field. This means that most recent methods and heuristics are not supported by mathematical theory. The field should therefore re-focus on theoretical issues such as problem analysis and convergence properties of new methods.

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Notes

  1. Discrete member-size Minimum weight Truss problem

  2. Discrete member-size Minimum weight Truss problem with Deflection constraints

  3. Note that this word is often used in a very broad sense in this article.

  4. The notation A≽B means that the matrix A−B is positive semidefinite.

  5. In this article only single load problems are stated to simplify the notation. All problem formulations, and reformulations, and most methods and heuristics can be generalized to multiple load cases.

  6. The notation mixed integer is herein used for optimization problems with both continuous and discrete/integer/0-1 variables. The term mixed discrete is often used in the structural optimization community.

  7. 7 Semidefinite Programming deals with optimization problems with matrix variables that must be symmetric and positive semidefinite or problems with linear matrix inequalities.

  8. This is, in fact, the predominant way of dealing with constraints in meta heuristics for optimal truss design.

  9. This section is an extension of parts of the Introduction in the doctoral thesis Stolpe (2013). Parts of this section are therefore identical to the corresponding paragraphs in Stolpe (2013).

  10. It can perhaps provide some consolation to those that feel targeted in this section that the author is occasionally throwing bricks in glass houses.

  11. This number does not include different variants and flavours within each class of heuristic or method.

  12. Development and dissemination of the benchmark library together with approval of new problem instances could be a task for a working group under the International Society for Structural and Multidisciplinary Optimization (ISSMO).

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Acknowledgments

I thank my colleague Susana Rojas-Labanda for providing many constructive suggestions and for proof-reading drafts of this manuscript. I would also like to sincerely thank three reviewers for detailed and constructive comments and suggestions which improved the article.

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  1. Department of Wind Energy, Technical University of Denmark (DTU), Frederiksborgvej 399, 4000, Roskilde, Denmark

    Mathias Stolpe

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Stolpe, M. Truss optimization with discrete design variables: a critical review. Struct Multidisc Optim 53, 349–374 (2016). https://doi.org/10.1007/s00158-015-1333-x

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