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On tackling reverse convex constraints for non-overlapping of unequal circles

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Abstract

We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications. The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield unsatisfactory loose relaxations. Consequently, solving such non-convex models to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we apply a purpose-built branching scheme on non-overlapping constraints and utilize strengthened intersection cuts and various feasibility-based tightening techniques to further tighten the model relaxation. We embed these techniques into a branch-and-bound code and test them on two variants of circle packing problems. Our computational studies on a suite of 75 benchmark instances yielded, for the first time in the open literature, a total of 54 provably optimal solutions, and it was demonstrated to be competitive over the use of the state-of-the-art general-purpose global optimization solvers.

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Notes

  1. For ease of exposition, we simplify the notation of Eq. (1), as follows: \(x_{i} \leftarrow a_{i^\prime } - a_{j^\prime }\), \(x_{j} \leftarrow b_{i^\prime } - b_{j^\prime }\) and \(\rho _{ij} \leftarrow r_{i^\prime } + r_{j^\prime }\), where \((a_{\ell }, b_{\ell })\) and \(r_{\ell }\) represent the center coordinates and radii, respectively, of two circles \(\ell \in \left\{ i^\prime , j^\prime \right\} \).

  2. For simplicity, we use here a common tolerance \(\varepsilon \) for all constraints in problem (2), but we remark that, in principle, one may use a set of constraint-specific feasibility tolerances that are appropriately scaled for each constraint.

  3. Structural variables x are not necessarily bounded, i.e., \(x^L_i \rightarrow -\infty \) and/or \(x_i^U \rightarrow +\infty \), for any and all \(i \in \{1, 2,\ldots , n\}\).

  4. From our computational experience, the number of vertices of \(FR_{ij}\) is usually small (around 10), and thus identifying its concave envelope via enumeration is computationally efficient. Regardless, if the number of vertices of \(FR_{ij}\) is large, one can always properly relax the domain \(FR_{ij}\) and obtain a new bounded polyhedron that contains \(FR_{ij}\) and has a small number of vertices.

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Acknowledgements

Akang Wang gratefully acknowledges financial support from the James C. Meade Graduate Fellowship and the H. William and Ruth Hamilton Prengle Graduate Fellowship at Carnegie Mellon University.

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Authors and Affiliations

  1. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Akang Wang & Chrysanthos E. Gounaris

  2. Center for Advanced Process Decision-making, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Akang Wang & Chrysanthos E. Gounaris

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  1. Akang Wang
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Correspondence to Chrysanthos E. Gounaris.

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Wang, A., Gounaris, C.E. On tackling reverse convex constraints for non-overlapping of unequal circles. J Glob Optim 80, 357–385 (2021). https://doi.org/10.1007/s10898-020-00976-y

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