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

Author

Listed:
  • Dominique Monnet

    (Polytechnique Montréal)

  • Warren Hare

    (University of British Columbia Okanagan)

  • Yves Lucet

    (University of British Columbia Okanagan)

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.

Suggested Citation

  • Dominique Monnet & Warren Hare & Yves Lucet, 2022. "A successive relaxation algorithm to solve a MILP involving piecewise linear functions with application to road design," Computational Optimization and Applications, Springer, vol. 81(3), pages 741-767, April.
    • Handle: RePEc:spr:coopap:v:81:y:2022:i:3:d:10.1007_s10589-021-00347-7
    DOI: 10.1007/s10589-021-00347-7
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    References listed on IDEAS

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    1. Dominique Monnet & Warren Hare & Yves Lucet, 2020. "Fast feasibility check of the multi-material vertical alignment problem in road design," Computational Optimization and Applications, Springer, vol. 75(2), pages 515-536, March.
    2. Moreb, Ahmad A., 1996. "Linear programming model for finding optimal roadway grades that minimize earthwork cost," European Journal of Operational Research, Elsevier, vol. 93(1), pages 148-154, August.
    3. Hare, Warren L. & Koch, Valentin R. & Lucet, Yves, 2011. "Models and algorithms to improve earthwork operations in road design using mixed integer linear programming," European Journal of Operational Research, Elsevier, vol. 215(2), pages 470-480, December.
    4. Hare, Warren & Lucet, Yves & Rahman, Faisal, 2015. "A mixed-integer linear programming model to optimize the vertical alignment considering blocks and side-slopes in road construction," European Journal of Operational Research, Elsevier, vol. 241(3), pages 631-641.
    5. Ahmet B. Keha & Ismael R. de Farias & George L. Nemhauser, 2006. "A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization," Operations Research, INFORMS, vol. 54(5), pages 847-858, October.
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