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A hybrid method for multidimensional scaling using city-block distances

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  • Antanas Žilinskas
  • Julius Žilinskas

Abstract

The problem of multidimensional scaling with city-block distances in the embedding space is reduced to a two level optimization problem consisting of a combinatorial problem at the upper level and a quadratic programming problem at the lower level. A hybrid method is proposed combining randomized search for the upper level problem with a standard quadratic programming algorithm for the lower level problem. Several algorithms for the combinatorial problem have been tested and an evolutionary global search algorithm has been proved most suitable. An experimental code of the proposed hybrid multidimensional scaling algorithm is developed and tested using several test problems of two- and three-dimensional scaling. Copyright Springer-Verlag 2008

Suggested Citation

  • Antanas Žilinskas & Julius Žilinskas, 2008. "A hybrid method for multidimensional scaling using city-block distances," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 429-443, December.
    • Handle: RePEc:spr:mathme:v:68:y:2008:i:3:p:429-443
    DOI: 10.1007/s00186-008-0238-5
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    References listed on IDEAS

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    1. Phipps Arabie, 1991. "Was euclid an unnecessarily sophisticated psychologist?," Psychometrika, Springer;The Psychometric Society, vol. 56(4), pages 567-587, December.
    2. 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.
    3. 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;The Classification Society, vol. 16(2), pages 225-254, July.
    4. Alex Murillo & J. Fernando Vera & Willem J. Heiser, 2005. "A Permutation-Translation Simulated Annealing Algorithm for L 1 and L 2 Unidimensional Scaling," Journal of Classification, Springer;The Classification Society, vol. 22(1), pages 119-138, June.
    5. Patrick Groenen & Rudolf Mathar & Willem Heiser, 1995. "The majorization approach to multidimensional scaling for Minkowski distances," Journal of Classification, Springer;The Classification Society, vol. 12(1), pages 3-19, March.
    6. Lawrence Hubert & Phipps Arabie & Matthew Hesson-Mcinnis, 1992. "Multidimensional scaling in the city-block metric: A combinatorial approach," Journal of Classification, Springer;The Classification Society, vol. 9(2), pages 211-236, December.
    7. Jan Leeuw, 1984. "Differentiability of Kruskal's stress at a local minimum," Psychometrika, Springer;The Psychometric Society, vol. 49(1), pages 111-113, March.
    8. Michael J. Brusco, 2001. "A Simulated Annealing Heuristic for Unidimensional and Multidimensional (City-Block) Scaling of Symmetric Proximity Matrices," Journal of Classification, Springer;The Classification Society, vol. 18(1), pages 3-33, January.
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    Cited by:

    1. Julius Žilinskas, 2012. "Parallel branch and bound for multidimensional scaling with city-block distances," Journal of Global Optimization, Springer, vol. 54(2), pages 261-274, October.

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