A new Branch and Bound method for a discrete truss topology design problem

Abstract

Our paper considers a classic problem in the field of Truss Topology Design, the goal of which is to determine the stiffest truss, under a given load, with a bound on the total volume and discrete requirements in the cross-sectional areas of the bars. To solve this problem we propose a new two-stage Branch and Bound algorithm.

In the first stage we perform a Branch and Bound algorithm on the nodes of the structure. This is based on the following dichotomy study: either a node is in the final structure or not. In the second stage, a Branch and Bound on the bar areas is conducted. The existence or otherwise of a node in this structure is ensured by adding constraints on the cross-sectional areas of its incident bars. In practice, for reasons of stability, free bars linked at free nodes should be avoided. Therefore, if a node exists in the structure, then there must be at least two incident bars on it, unless it is a supported node. Thus, a new constraint is added, which lower bounds the sum of the cross-sectional areas of bars incident to the node. Otherwise, if a free node does not belong to the final structure, then all the bar area variables corresponding to bars incident to this node may be set to zero. These constraints are added during the first stage and lead to a tight model. We report the computational experiments conducted to test the effectiveness of this two-stage approach, enhanced by the rule to prevent free bars, as compared to a classical Branch and Bound algorithm, where branching is only performed on the bar areas.

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Algorithm 1
Fig. 3

Notes

  1. 1.

    In the objective function, to simplify, we consider twice the compliance.

  2. 2.

    \(|\mathcal{A}|\) denotes the cardinal of set \(\mathcal{A}\).

  3. 3.

    http://www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-Suite.aspx.

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Acknowledgements

This work is supported by National Funding from FCT—Fundação para a Ciência e a Tecnologia, under the project: PEst-OE/MAT/UI0152.

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Correspondence to Adelaide Cerveira.

Appendix: Convex quadratically constrained problem

Appendix: Convex quadratically constrained problem

Theorem 1

Consider the following non-convex problem

and consider the following quadratically constrained minimization problem

with \(\underline{V}_{i}=L_{i} s_{i}\) and \(\overline{V}_{i}=U_{i}s_{i}\), i=1,…,m. Similarly to Theorem 4.1 in [4], the following holds:

  1. (a)

    In all cases, whether \((\overline{R})\) or (Q2P) possess a solution or not, we have the identity \(\mathit{inf}(\overline{R})=-\mathit{inf}(\mathit{Q2P})\), where \(\mathit{inf}(\overline{R})=-\infty\) (inf(Q2P)=−∞) if \(\overline{R}\) (resp. Q2P) is infeasible and \(\mathit{inf}(\overline{R})=\min(\overline{R})\) (resp. \(\mathit{inf} (\mathit{Q2P})=\min(\overline{R})\)) otherwise.

  2. (b)

    Let (u,r,γ,λ) be optimal for (Q2P) with corresponding Lagrange-multipliers \(\tau^{-}, \tau^{+} \in \mathbb{R}^{m}_{+}\). Then (a,u) is optimal for \((\overline{R})\) where

    $$a_i = \tau^-\underline{V}_i+ \tau^+ \overline{V}_i \qquad \forall i= 1,\ldots,m. $$
  3. (c)

    The following statements are equivalent:

    1. (c1)

      Problem \((\overline{R})\) possesses a solution.

    2. (c2)

      Problem \((\overline{R})\) is feasible.

    3. (c3)

      Problem (Q2P) is bounded.

    4. (c4)

      Problem (Q2P) possesses a solution.

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Cerveira, A., Agra, A., Bastos, F. et al. A new Branch and Bound method for a discrete truss topology design problem. Comput Optim Appl 54, 163–187 (2013). https://doi.org/10.1007/s10589-012-9487-6

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Keywords

  • Truss topology design
  • Stiffness
  • Semi-definite programming
  • Quadratic programming
  • Global optimization
  • Branch and Bound