Parallel Global Optimization in Multidimensional Scaling

  • Julius Žilinskas
Part of the Springer Optimization and Its Applications book series (SOIA, volume 27)


Multidimensional scaling is a technique for exploratory analysis of multidimensional data, whose essential part is optimization of a function possessing many adverse properties including multidimensionality, multimodality, and non-differentiability. In this chapter, global optimization algorithms for multidimensional scaling are reviewed with particular emphasis on parallel computing.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arabie, P.: Was Euclid an unnecessarily sophisticated psychologist? Psychometrika 56(4), 567–587 (1991). doi 10.1007/BF02294491zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baravykaitė, M., Čiegis, R.: An implementation of a parallel generalized branch and bound template. Mathematical Modelling and Analysis 12(3), 277–289 (2007) DOI 10.3846/1392- 6292.2007.12.277-289zbMATHCrossRefGoogle Scholar
  3. 3.
    Baravykaitė, M., Čiegis, R., Žilinskas, J.: Template realization of generalized branch and bound algorithm. Mathematical Modelling and Analysis 10(3), 217–236 (2005)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Baravykaitė, M., Žilinskas, J.: Implementation of parallel optimization algorithms using generalized branch and bound template. In: I.D.L. Bogle, J. Žilinskas (eds.) Computer Aided Methods in Optimal Design and Operations, Series on Computers and Operations Research, vol. 7, pp. 21–28. World Scientific, Singapore (2006)CrossRefGoogle Scholar
  5. 5.
    Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications, 2nd edn. Springer, New York (2005)zbMATHGoogle Scholar
  6. 6.
    Brusco, M.J.: A simulated annealing heuristic for unidimensional and multidimensional (city-block) scaling of symmetric proximity matrices. Journal of Classification 18(1), 3–33 (2001)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Cantú-Paz, E.: Efficient and Accurate Parallel Genetic Algorithms. Kluwer Academic Publishers, New York (2000)Google Scholar
  8. 8.
    Černý, V.: Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications 45(1), 41–51 (1985). doi 10.1007/BF00940812zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, 2nd edn. Chapman & Hall/CRC, Boca Raton (2001)zbMATHGoogle Scholar
  10. 10.
    Floudas, C.A.: Deterministic Global Optimization: Theory, Methods and Applications, Nonconvex Optimization and its Applications, vol. 37. Kluwer Academic Publishers, New York (2000)Google Scholar
  11. 11.
    Glover, F.: Tabu search – Part I. ORSA Journal on Computing 1(3), 190–206 (1989)zbMATHGoogle Scholar
  12. 12.
    Glover, F.: Tabu search – Part II. ORSA Journal on Computing 2(1), 4–32 (1990)zbMATHGoogle Scholar
  13. 13.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Adison-Wesley, Reading, MA (1989)Google Scholar
  14. 14.
    Green, P., Carmone, F., Smith, S.: Multidimensional Scaling: Concepts and Applications. Allyn and Bacon, Boston (1989)Google Scholar
  15. 15.
    Groenen, P., Mathar, R., Trejos, J.: Global optimization methods for multidimensional scaling applied to mobile communication. In: W. Gaul, O. Opitz, M. Schander (eds.) Data Analysis: Scientific Modeling and Practical Applications, pp. 459–475. Springer, New York (2000)Google Scholar
  16. 16.
    Groenen, P.J.F.: The Majorization Approach to Multidimentional Scaling: Some Problems and Extensions. DSWO Press, Leiden (1993)Google Scholar
  17. 17.
    Groenen, P.J.F., Heiser, W.J.: The tunneling method for global optimization in multidimensional scaling. Psychometrika 61(3), 529–550 (1996). doi 10.1007/BF02294553zbMATHCrossRefGoogle Scholar
  18. 18.
    Groenen, P.J.F., Mathar, R., Heiser, W.J.: The majorization approach to multidimensional scaling for Minkowski distances. Journal of Classification 12(1), 3–19 (1995). doi 10.1007/BF01202265zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hansen, E., Walster, G.W.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New York (2003)Google Scholar
  20. 20.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, Nonconvex Optimization and its Applications, vol. 48, 2nd edn. Kluwer Academic Publishers, New York (2001)Google Scholar
  21. 21.
    Kirkpatrick, S., Gelatt, C.D.J., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983). doi 10.1126/science.220.4598.671CrossRefMathSciNetGoogle Scholar
  22. 22.
    Leeuw, J.D.: Differentiability of Kruskal’s stress at a local minimum. Psychometrika 49(1), 111–113 (1984). doi 10.1007/BF02294209CrossRefMathSciNetGoogle Scholar
  23. 23.
    Mathar, R.: A hybrid global optimization algorithm for multidimensional scaling. In: R. Klar, O. Opitz (eds.) Classification and Knowledge Organization, pp. 63–71. Springer, New York (1997)Google Scholar
  24. 24.
    Mathar, R., Žilinskas, A.: On global optimization in two-dimensional scaling. Acta Applicandae Mathematicae 33(1), 109–118 (1993). doi 10.1007/BF00995497zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Michalewich, Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer, Berlin (1996)Google Scholar
  26. 26.
    Mockus, J.: Bayesian Approach to Global Optimization. Kluwer Academic Publishers, Boston (1989)Google Scholar
  27. 27.
    Schwefel, H.P.: Evolution and Optimum Seeking. John Wiley & Sons, New York (1995)Google Scholar
  28. 28.
    Törn, A., Žilinskas, A.: Global optimization. Lecture Notes in Computer Science 350, 1–252 Springer-Verlag, Berlin (1989). doi 10.1007/3-540-50871-6Google Scholar
  29. 29.
    Varoneckas, A., Žilinskas, A., Žilinskas, J.: Multidimensional scaling using parallel genetic algorithm. In: I.D.L. Bogle, J. Žilinskas (eds.) Computer Aided Methods in Optimal Design and Operations, Series on Computers and Operations Research, vol. 7, pp. 129–138. World Scientific, Singapore (2006)CrossRefGoogle Scholar
  30. 30.
    Vera, J.F., Heiser, W.J., Murillo, A.: Global optimization in any Minkowski metric: a permutation-translation simulated annealing algorithm for multidimensional scaling. Journal of Classification 24(2), 277–301 (2007). doi 10.1007/s00357-007-0020-1CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Žilinskas, A., Žilinskas, J.: Parallel hybrid algorithm for global optimization of problems occurring in MDS-based visualization. Computers & Mathematics with Applications 52(1-2), 211–224 (2006). doi 10.1016/j.camwa.2006.08.016zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Žilinskas, A., Žilinskas, J.: Parallel genetic algorithm: assessment of performance in multidimensional scaling. In: GECCO ’07: Proceedings of the 9th annual conference on Genetic and evolutionary computation, pp. 1492–1501. ACM, New York (2007). doi 10.1145/1276958.1277229Google Scholar
  33. 33.
    Žilinskas, A., Žilinskas, J.: Two level minimization in multidimensional scaling. Journal of Global Optimization 38(4), 581–596 (2007). doi 10.1007/s10898-006-9097-xzbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Žilinskas, A., Žilinskas, J.: Branch and bound algorithm for multidimensional scaling with city-block metric. Journal of Global Optimization in press (2008). doi 10.1007/s10898-008-9306-xGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Julius Žilinskas
    • 1
  1. 1.Institute of Mathematics and InformaticsLithuania

Personalised recommendations