Skip to main content
SpringerLink
Log in

Parallel Global Optimization in Multidimensional Scaling

  • Chapter
Parallel Scientific Computing and Optimization

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

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arabie, P.: Was Euclid an unnecessarily sophisticated psychologist? Psychometrika 56(4), 567–587 (1991). doi 10.1007/BF02294491

    Article  MATH  MathSciNet  Google Scholar 

  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- 6292.2007.12.277-289

    Article  MATH  Google Scholar 

  3. Baravykaitė, M., Čiegis, R., Žilinskas, J.: Template realization of generalized branch and bound algorithm. Mathematical Modelling and Analysis 10(3), 217–236 (2005)

    MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  5. Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications, 2nd edn. Springer, New York (2005)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  7. Cantú-Paz, E.: Efficient and Accurate Parallel Genetic Algorithms. Kluwer Academic Publishers, New York (2000)

    Google Scholar 

  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/BF00940812

    Article  MATH  MathSciNet  Google Scholar 

  9. Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, 2nd edn. Chapman & Hall/CRC, Boca Raton (2001)

    MATH  Google Scholar 

  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. Glover, F.: Tabu search – Part I. ORSA Journal on Computing 1(3), 190–206 (1989)

    MATH  Google Scholar 

  12. Glover, F.: Tabu search – Part II. ORSA Journal on Computing 2(1), 4–32 (1990)

    MATH  Google Scholar 

  13. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Adison-Wesley, Reading, MA (1989)

    Google Scholar 

  14. Green, P., Carmone, F., Smith, S.: Multidimensional Scaling: Concepts and Applications. Allyn and Bacon, Boston (1989)

    Google Scholar 

  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. Groenen, P.J.F.: The Majorization Approach to Multidimentional Scaling: Some Problems and Extensions. DSWO Press, Leiden (1993)

    Google Scholar 

  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/BF02294553

    Article  MATH  Google Scholar 

  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/BF01202265

    Article  MATH  MathSciNet  Google Scholar 

  19. Hansen, E., Walster, G.W.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New York (2003)

    Google Scholar 

  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. Kirkpatrick, S., Gelatt, C.D.J., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983). doi 10.1126/science.220.4598.671

    Article  MathSciNet  Google Scholar 

  22. Leeuw, J.D.: Differentiability of Kruskal’s stress at a local minimum. Psychometrika 49(1), 111–113 (1984). doi 10.1007/BF02294209

    Article  MathSciNet  Google Scholar 

  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. Mathar, R., Žilinskas, A.: On global optimization in two-dimensional scaling. Acta Applicandae Mathematicae 33(1), 109–118 (1993). doi 10.1007/BF00995497

    Article  MATH  MathSciNet  Google Scholar 

  25. Michalewich, Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer, Berlin (1996)

    Google Scholar 

  26. Mockus, J.: Bayesian Approach to Global Optimization. Kluwer Academic Publishers, Boston (1989)

    Google Scholar 

  27. Schwefel, H.P.: Evolution and Optimum Seeking. John Wiley & Sons, New York (1995)

    Google Scholar 

  28. Törn, A., Žilinskas, A.: Global optimization. Lecture Notes in Computer Science 350, 1–252 Springer-Verlag, Berlin (1989). doi 10.1007/3-540-50871-6

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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/s00357-007-0020-1

    Article  MATH  MathSciNet  Google Scholar 

  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.2006.08.016

    Article  MATH  MathSciNet  Google Scholar 

  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 10.1145/1276958.1277229

    Google Scholar 

  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-x

    Article  MATH  MathSciNet  Google Scholar 

  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-x

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania

    Julius Žilinskas

Authors
  1. Julius Žilinskas
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Žilinskas, J. (2009). Parallel Global Optimization in Multidimensional Scaling. In: Parallel Scientific Computing and Optimization. Springer Optimization and Its Applications, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09707-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation