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A successive relaxation algorithm to solve a MILP involving piecewise linear functions with application to road design

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Abstract

This paper presents a new algorithm to build feasible solutions to a MILP formulation of the vertical alignment problem in road design. This MILP involves a large number of special ordered set of type 2 variables used to describe piecewise linear functions. The principle of the algorithm is to successively solve LPs adapted from the MILP by replacing the special ordered set of type 2 constraints by linear constraints. Proof that the solutions to the successive linear relaxations of the MILP converge to a feasible solution to the MILP is provided. Numerical results emphasize that the algorithm performs better than CPLEX   for large scale vertical alignment problems.

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Data availability

The datasets generated and analyzed during the current study are subject to a collaborative research agreement between UBC and Softree Technical Systems Inc. Interested parties should contact Softree Technical Systems Inc. directly to request access to those datasets.

Notes

  1. https://www.ibm.com/docs/en/icos/12.8.0.0?topic=cplex-list-parameters (28/05/2021).

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Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through Collaborative Research and Development grant #CRDPJ 479316-15 sponsored by Softree Technical Systems Inc. Part of the computation in this research was carried out using a software library provided by Softree Technical System Inc. Part of the research was performed in the Computer-Aided Convex Analysis (CA2) laboratory funded by a Leaders Opportunity Fund (LOF, John R. Evans Leaders Fund, Funding for research infrastructure) from the Canada Foundation for Innovation (CFI) and by a British Columbia Knowledge Development Fund (BCKDF).

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  1. Dominique Monnet

    Present address: Polytechnique Montréal, Montréal, Canada

Authors and Affiliations

  1. University of British Columbia Okanagan, Kelowna, Canada

    Warren Hare & Yves Lucet

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  1. Dominique Monnet
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  2. Warren Hare
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  3. Yves Lucet
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Correspondence to Dominique Monnet.

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Appendix: Additional performance profiles

Appendix: Additional performance profiles

See Figs. 5 and 6

Fig. 5
figure 5

Performance profiles of BHLH17   and \(P_2T_{0.05}\) for 1% and 5% gaps over the sets of roads involving less and more than 20 vertical offsets (number of discretization points for volume approximation). The x axis in Figure (d) is in logarithmic scale

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Fig. 6
figure 6

Performance profiles of BHLH17   and \(P_2T_{0.05}\) for 1% and 5% gaps over the sets of roads involving less and more than 450 sections

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Monnet, D., Hare, W. & Lucet, Y. A successive relaxation algorithm to solve a MILP involving piecewise linear functions with application to road design. Comput Optim Appl 81, 741–767 (2022). https://doi.org/10.1007/s10589-021-00347-7

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