Encyclopedia of Optimization

2009 Edition
| Editors: Christodoulos A. Floudas, Panos M. Pardalos

Optimization-Based Visualization

  • Antanas Žilinskas
  • Julius Žilinskas
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-74759-0_478

Article Outline

Keywords and Phrases




See also


Keywords and Phrases

Global optimization Unidimensional scaling Multidimensional scaling 
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  1. 1.
    Arabie P (1991) Was Euclid an unnecessarily sophisticated psychologist? Psychometrika 56:567–587Google Scholar
  2. 2.
    Borg I, Groenen PJF (2005) Modern Multidimensional Scaling: Theory and Applications, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  3. 3.
    Brusco MJ (2001) A simulated annealing heuristic for unidimensional and multidimensional (city-block) scaling of symmetric proximity matrices. J Classif 18:3–33MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brusco MJ (2002) Integer programming methods for seriation and unidimensional scaling of proximity matrices: a review and some extensions. J Classif 19:45–67MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brusco MJ (2006) On the performance of simulated annealing for large-scale L 2 unidimensional scaling. J Classif 23:255–268MathSciNetGoogle Scholar
  6. 6.
    Brusco MJ, Stahl S (2000) Using quadratic assignment methods to generate initial permutations for least-squares unidimensional scaling of symmetric proximity matrices. J Classif 17:197–223MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brusco MJ, Stahl S (2005) Optimal least-squares unidimensional scaling: improved branch-and-bound procedures and comparison to dynamic programming. Psychometrika 70:253–270MathSciNetGoogle Scholar
  8. 8.
    Chen C (2003) Mapping Scientific Frontiers: The Quest for Knowledge Visualization. Springer, LondonGoogle Scholar
  9. 9.
    Cox TF, Cox MAA (2001) Multidimensional Scaling, 2nd edn. Chapman & Hall/CRC, Boca RatonzbMATHGoogle Scholar
  10. 10.
    Defays D (1978) A short note on a method of seriation. Br J Math Statist Psychol 31:49–53Google Scholar
  11. 11.
    de Leeuw J (1977) Applications of convex analysis to multidimensional scaling. In: Barra JR, Brodeau F, Romier G, van Cutsem B (eds) Recent Developments in Statistics. North-Holland, Amsterdam, pp 133–145Google Scholar
  12. 12.
    de Leeuw J (1984) Differentiability of Kruskal's stress at a local minimum. Psychometrika 49:111–113MathSciNetGoogle Scholar
  13. 13.
    de Leeuw J (1988) Convergence of the majorization method for multidimensional scaling. J Classif 5:163–180zbMATHGoogle Scholar
  14. 14.
    de Leeuw J, Heiser W (1982) Theory of multidimensional scaling. Handb Stat 2:285–316Google Scholar
  15. 15.
    DeSarbo WS, Kim Y, Wedel M, Fong DKH (1998) A Bayesian approach to the spatial representation of market structure from consumer choice data. Eur J Oper Res 111:285–305zbMATHGoogle Scholar
  16. 16.
    Everett JE (2001) Algorithms for multidimensional scaling. In: Chambers LD (ed) The Practical Handbook of Genetic Algorithms: Applications, 2nd edn. Chapman & Hall/CRC, Boca Raton, pp 203–233Google Scholar
  17. 17.
    Groenen PJF (1993) The Majorization Approach to Multidimensional Scaling: Some Problems and Extensions. DSWO Press, LeidenzbMATHGoogle Scholar
  18. 18.
    Groenen PJF, Heiser WJ (1996) The tunneling method for global optimization in multidimensional scaling. Psychometrika 61:529–550zbMATHGoogle Scholar
  19. 19.
    Groenen PJF, Heiser WJ, Meulman JJ (1998) City-block scaling: smoothing strategies for avoiding local minima. In: Balderjahn I, Mathar R, Schader M (eds) Classification, Data Analysis, and Data Highways. Springer, Berlin, pp 46–53Google Scholar
  20. 20.
    Groenen PJF, Heiser WJ, Meulman JJ (1999) Global optimization in least-squares multidimensional scaling by distance smoothing. J Classif 16:225–254MathSciNetzbMATHGoogle Scholar
  21. 21.
    Groenen PJF, Mathar R, Heiser WJ (1995) The majorization approach to multidimensional scaling for Minkowski distances. J Classif 12:3–19MathSciNetzbMATHGoogle Scholar
  22. 22.
    Groenen P, Mathar R, Trejos J (2000) Global optimization methods for multidimensional scaling applied to mobile communication. In: Gaul W, Opitz O, Schander M (eds) Data Analysis: Scientific Modeling and Practical Applications. Springer, Berlin, pp 459–475Google Scholar
  23. 23.
    Hubert LJ, Golledge RG (1981) Matrix reorganization and dynamic programming: applications to paired comparisons and unidimensional seriation. Psychometrika 46:429–441zbMATHGoogle Scholar
  24. 24.
    Hubert LJ, Arabie P, Meulman JJ (2002) Linear unidimensional scaling in the L 2-norm: basic optimization methods using MATLAB. J Classif 19:303–328MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kearsley AJ, Tapia RA, Trosset MW (1998) The solution of the metric STRESS and SSTRESS problems in multidimensional scaling using Newton's method. Comput Statist 13:369–396zbMATHGoogle Scholar
  26. 26.
    Klock H, Buhmann JM (2000) Data visualization by multidimensional scaling: a deterministic annealing approach. Pattern Recognit 33:651–669Google Scholar
  27. 27.
    Kruskal JB, Wish M (1978) Multidimensional Scaling. Bell Laboratories, Murray HillGoogle Scholar
  28. 28.
    Lau K, Leung PL, Tse K (1998) A nonlinear programming approach to metric unidimensional scaling. J Classif 15:3–14zbMATHGoogle Scholar
  29. 29.
    Leung PL, Lau K (2004) Estimating the city-block two-dimensional scaling model with simulated annealing. Eur J Oper Res 158:518–524zbMATHGoogle Scholar
  30. 30.
    Mathar R (1997) A hybrid global optimization algorithm for multidimensional scaling. In: Klar R, Opitz O (eds) Classification and Knowledge Organization. Springer, Berlin, pp 63–71Google Scholar
  31. 31.
    Mathar R (1997) Multidimensionale Skalierung, Mathematische Grundlagen und Algorithmische Konzepte. Teubner, LeipzigGoogle Scholar
  32. 32.
    Mathar R, Žilinskas A (1993) On global optimization in two-dimensional scaling. Acta Applicandae Mathematicae 33:109–118MathSciNetzbMATHGoogle Scholar
  33. 33.
    McIver JP, Carmines EG (1981) Unidimensional Scaling, Sage Publications, Newbury ParkGoogle Scholar
  34. 34.
    Miyano H, Inukai Y (1982) Sequential estimation in multidimensional scaling. Psychometrika 47:321–336MathSciNetGoogle Scholar
  35. 35.
    Murillo A, Vera JF, Heiser WJ (2005) A permutation-translation simulated annealing algorithm for L 1 and L 2 unidimensional scaling. J Classif 22:119–138MathSciNetzbMATHGoogle Scholar
  36. 36.
    Nelson TR, Rabianski J (1988) Consumer preferences in housing market analysis: an application of multidimensional scaling techniques. Real Estate Econ 16:138–159Google Scholar
  37. 37.
    Pliner V (1996) Metric unidimensional scaling and global optimization. J Classif 13:3–18MathSciNetzbMATHGoogle Scholar
  38. 38.
    Poole KT (1990) Least squares metric, unidimensional scaling of multivariate linear models. Psychometrika 55:123–149MathSciNetzbMATHGoogle Scholar
  39. 39.
    Schiffman SS, Reynolds ML, Young FW (1981) Introduction to Multidimensional Scaling: Theory, Methods, and Applications. Academic Press, LondonzbMATHGoogle Scholar
  40. 40.
    Simantiraki E (1996) Unidimensional scaling: a linear programming approach minimizing absolute deviations. J Classif 13:19–25zbMATHGoogle Scholar
  41. 41.
    Takane Y (2006) Applications of multidimensional scaling in psychometrics. Handb Stat 26:359–400Google Scholar
  42. 42.
    Torgerson WS (1958) Theory and Methods of Scaling. Wiley, New YorkGoogle Scholar
  43. 43.
    Žilinskas A, Fraga ES, Mackutė A (2006) Data analysis and visualisation for robust multi-criteria process optimisation. Comput Chem Eng 30:1061–1071Google Scholar
  44. 44.
    Žilinskas A, Žilinskas J (2007) Two level minimization in multidimensional scaling. J Global Optim 38:581–596MathSciNetzbMATHGoogle Scholar
  45. 45.
    Žilinskas J (2006) Multidimensional scaling in protein and pharmacological sciences. In: Bogle IDL, Žilinskas J (eds) Computer Aided Methods in Optimal Design and Operations. World Scientific, Singapore, pp 139–148Google Scholar

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© Springer-Verlag 2008

Authors and Affiliations

  • Antanas Žilinskas
    • 1
  • Julius Žilinskas
    • 1
  1. 1.Institute of Mathematics and InformaticsVilniusLithuania