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A specialized branch-and-bound algorithm for the Euclidean Steiner tree problem in n-space

Author

Listed:
  • Marcia Fampa

    (Universidade Federal do Rio de Janeiro)

  • Jon Lee

    (University of Michigan)

  • Wendel Melo

    (PESC/COPPE, Universidade Federal do Rio de Janeiro)

Abstract

We present a specialized branch-and-bound (b&b) algorithm for the Euclidean Steiner tree problem (ESTP) in $$\mathbb {R}^n$$ R n and apply it to a convex mixed-integer nonlinear programming (MINLP) formulation of the problem, presented by Fampa and Maculan. The algorithm contains procedures to avoid difficulties observed when applying a b&b algorithm for general MINLP problems to solve the ESTP. Our main emphasis is on isomorphism pruning, in order to prevent solving several equivalent subproblems corresponding to isomorphic Steiner trees. We introduce the concept of representative Steiner trees, which allows the pruning of these subproblems, as well as the implementation of procedures to fix variables and add valid inequalities. We also propose more general procedures to improve the efficiency of the b&b algorithm, which may be extended to the solution of other MINLP problems. Computational results demonstrate substantial gains compared to the standard b&b for convex MINLP.

Suggested Citation

  • Marcia Fampa & Jon Lee & Wendel Melo, 2016. "A specialized branch-and-bound algorithm for the Euclidean Steiner tree problem in n-space," Computational Optimization and Applications, Springer, vol. 65(1), pages 47-71, September.
    • Handle: RePEc:spr:coopap:v:65:y:2016:i:1:d:10.1007_s10589-016-9835-z
    DOI: 10.1007/s10589-016-9835-z
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    References listed on IDEAS

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    1. Omprakash K. Gupta & A. Ravindran, 1985. "Branch and Bound Experiments in Convex Nonlinear Integer Programming," Management Science, INFORMS, vol. 31(12), pages 1533-1546, December.
    2. Wendel Melo & Marcia Fampa & Fernanda Raupp, 2014. "Integrating nonlinear branch-and-bound and outer approximation for convex Mixed Integer Nonlinear Programming," Journal of Global Optimization, Springer, vol. 60(2), pages 373-389, October.
    3. Nelson Maculan & Philippe Michelon & Adilson Xavier, 2000. "The Euclidean Steiner tree problem in R n : A mathematical programming formulation," Annals of Operations Research, Springer, vol. 96(1), pages 209-220, November.
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    Cited by:

    1. Renan Vicente Pinto & Nelson Maculan, 2023. "A new heuristic for the Euclidean Steiner Tree Problem in $${\mathbb {R}}^n$$ R n," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 391-413, July.
    2. Wendel Melo & Marcia Fampa & Fernanda Raupp, 2020. "An overview of MINLP algorithms and their implementation in Muriqui Optimizer," Annals of Operations Research, Springer, vol. 286(1), pages 217-241, March.
    3. Hacene Ouzia & Nelson Maculan, 2022. "Mixed integer nonlinear optimization models for the Euclidean Steiner tree problem in $$\mathbb {R}^d$$ R d," Journal of Global Optimization, Springer, vol. 83(1), pages 119-136, May.

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