On Multidimensional Scaling with City-Block Distances

  • Nerijus Galiauskas
  • Julius ŽilinskasEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8426)


Multidimensional scaling is a technique for exploratory analysis of multidimensional data. The essential part of the technique is minimization of a function with unfavorable properties like multimodality, non-differentiability, and invariability with respect to some transformations. Recently various two-level optimization algorithms for multidimensional scaling with city-block distances have been proposed exploiting piecewise quadratic structure of the least squares objective function with such distances. A problem of combinatorial optimization is tackled at the upper level, and convex quadratic programming problems are tackled at the lower level. In this paper we discuss a new reformulation of the problem where lower level quadratic programming problems seem more suited for two-level optimization.


Multidimensional scaling City-block distances Multilevel optimization Global optimization 



This research was funded by a grant (No. MIP-063/2012) from the Research Council of Lithuania.


  1. 1.
    Arabie, P.: Was Euclid an unnecessarily sophisticated psychologist? Psychometrika 56(4), 567–587 (1991). doi: 10.1007/BF02294491 CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications, 2nd edn. Springer, New York (2005)Google Scholar
  3. 3.
    Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, 2nd edn. Chapman & Hall/CRC, Boca Raton (2001)zbMATHGoogle Scholar
  4. 4.
    de Leeuw, J.: Differentiability of Kruskal’s stress at a local minimum. Psychometrika 49(1), 111–113 (1984). doi: 10.1007/BF02294209 CrossRefMathSciNetGoogle Scholar
  5. 5.
    Dzemyda, G., Kurasova, O., Žilinskas, J.: Multidimensional Data Visualization: Methods and Applications. Springer, New York (2013). doi: 10.1007/978-1-4419-0236-8 CrossRefGoogle Scholar
  6. 6.
    Groenen, P.J.F., Mathar, R., Heiser, W.J.: The majorization approach to multidimensional scaling for Minkowski distances. J. Classif. 12(1), 3–19 (1995). doi: 10.1007/BF01202265 CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Redondo, J.L., Ortigosa, P.M., Žilinskas, J.: Multimodal evolutionary algorithm for multidimensional scaling with city-block distances. Informatica 23(4), 601–620 (2012)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Žilinskas, A., Žilinskas, J.: Two level minimization in multidimensional scaling. J. Global Optim. 38(4), 581–596 (2007). doi: 10.1007/s10898-006-9097-x CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Žilinskas, A., Žilinskas, J.: A hybrid method for multidimensional scaling using city-block distances. Math. Methods Oper. Res. 68(3), 429–443 (2008). doi: 10.1007/s00186-008-0238-5 CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Žilinskas, A., Žilinskas, J.: Branch and bound algorithm for multidimensional scaling with city-block metric. J. Global Optim. 43(2–3), 357–372 (2009). doi: 10.1007/s10898-008-9306-x CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Žilinskas, J.: Reducing of search space of multidimensional scaling problems with data exposing symmetries. Inf. Technol. Control 36(4), 377–382 (2007)Google Scholar
  12. 12.
    Žilinskas, J.: Parallel branch and bound for multidimensional scaling with city-block distances. J. Global Optim. 54(2), 261–274 (2012). doi: 10.1007/s10898-010-9624-7 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Vilnius University Institute of Mathematics and InformaticsVilniusLithuania

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