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The 2-opt behavior of the Hopfield Network applied to the TSP

Author

Listed:
  • Lucas García

    (Universidad Complutense de Madrid)

  • Pedro M. Talaván

    (Instituto Nacional de Estadística)

  • Javier Yáñez

    (Universidad Complutense de Madrid)

Abstract

The Continuous Hopfield Network (CHN) became one of the major breakthroughs in the come back of Neural Networks in the mid 80s, as it could be used to solve combinatorial optimization problems such as the Traveling Salesman Problem. Once researchers provided a mechanism, not based in trial-and-error, to guarantee the feasibility of the CHN, the quality of the solution was inferior to the ones provided by other heuristics. The next natural step is to study the behavior of the CHN as an optimizer, in order to improve its performance. With this regard, this paper analyzes the attractor basins of the CHN and establishes the mathematical foundations that guarantee the behavior of the network as a 2-opt; with the aim to open a new research line in which the CHN may be used, given the appropriate parameter setting, to solve a k-opt, which would make the network highly competitive. The analysis of the attraction basins of the CHN and its interpretation as a 2-opt is the subject of this article.

Suggested Citation

  • Lucas García & Pedro M. Talaván & Javier Yáñez, 2022. "The 2-opt behavior of the Hopfield Network applied to the TSP," Operational Research, Springer, vol. 22(2), pages 1127-1155, April.
    • Handle: RePEc:spr:operea:v:22:y:2022:i:2:d:10.1007_s12351-020-00585-3
    DOI: 10.1007/s12351-020-00585-3
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    References listed on IDEAS

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    1. Gerhard Reinelt, 1991. "TSPLIB—A Traveling Salesman Problem Library," INFORMS Journal on Computing, INFORMS, vol. 3(4), pages 376-384, November.
    2. S. Lin & B. W. Kernighan, 1973. "An Effective Heuristic Algorithm for the Traveling-Salesman Problem," Operations Research, INFORMS, vol. 21(2), pages 498-516, April.
    3. G. A. Croes, 1958. "A Method for Solving Traveling-Salesman Problems," Operations Research, INFORMS, vol. 6(6), pages 791-812, December.
    4. Helsgaun, Keld, 2000. "An effective implementation of the Lin-Kernighan traveling salesman heuristic," European Journal of Operational Research, Elsevier, vol. 126(1), pages 106-130, October.
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