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Parallel branch and bound for multidimensional scaling with city-block distances

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  • Julius Žilinskas

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    Abstract

    Multidimensional scaling is a technique for exploratory analysis of multidimensional data. The essential part of the technique is minimization of a multimodal function with unfavorable properties like invariants and non-differentiability. Recently a branch and bound algorithm for multidimensional scaling with city-block distances has been proposed for solution of medium-size problems exactly. The algorithm exploits piecewise quadratic structure of the objective function. In this paper a parallel version of the branch and bound algorithm for multidimensional scaling with city-block distances has been proposed and investigated. Parallel computing enabled solution of larger problems what was not feasible with the sequential version. Copyright Springer Science+Business Media, LLC. 2012

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    File URL: http://hdl.handle.net/10.1007/s10898-010-9624-7
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    Bibliographic Info

    Article provided by Springer in its journal Journal of Global Optimization.

    Volume (Year): 54 (2012)
    Issue (Month): 2 (October)
    Pages: 261-274

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    Handle: RePEc:spr:jglopt:v:54:y:2012:i:2:p:261-274

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    Web page: http://www.springer.com/business/operations+research/journal/10898

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    Related research

    Keywords: Multidimensional scaling; City-block distances; Multilevel optimization; Global optimization; Branch and bound; Parallel computing;

    References

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    1. Jan Leeuw, 1984. "Differentiability of Kruskal's stress at a local minimum," Psychometrika, Springer, vol. 49(1), pages 111-113, March.
    2. J. Fernando Vera & Willem J. Heiser & Alex Murillo, 2007. "Global Optimization in Any Minkowski Metric: A Permutation-Translation Simulated Annealing Algorithm for Multidimensional Scaling," Journal of Classification, Springer, vol. 24(2), pages 277-301, September.
    3. Patrick Groenen & Rudolf Mathar & Willem Heiser, 1995. "The majorization approach to multidimensional scaling for Minkowski distances," Journal of Classification, Springer, vol. 12(1), pages 3-19, March.
    4. Phipps Arabie, 1991. "Was euclid an unnecessarily sophisticated psychologist?," Psychometrika, Springer, vol. 56(4), pages 567-587, December.
    5. Leung, Pui Lam & Lau, Kin-nam, 2004. "Estimating the city-block two-dimensional scaling model with simulated annealing," European Journal of Operational Research, Elsevier, vol. 158(2), pages 518-524, October.
    6. P. J. F. Groenen & W. J. Heiser & J. J. Meulman, 1999. "Global Optimization in Least-Squares Multidimensional Scaling by Distance Smoothing," Journal of Classification, Springer, vol. 16(2), pages 225-254, July.
    7. Lawrence Hubert & Phipps Arabie & Matthew Hesson-Mcinnis, 1992. "Multidimensional scaling in the city-block metric: A combinatorial approach," Journal of Classification, Springer, vol. 9(2), pages 211-236, December.
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