Mathematical Programming

, Volume 169, Issue 1, pp 119–140 | Cite as

Visualizing data as objects by DC (difference of convex) optimization

  • Emilio Carrizosa
  • Vanesa Guerrero
  • Dolores Romero Morales
Full Length Paper Series B


In this paper we address the problem of visualizing in a bounded region a set of individuals, which has attached a dissimilarity measure and a statistical value, as convex objects. This problem, which extends the standard Multidimensional Scaling Analysis, is written as a global optimization problem whose objective is the difference of two convex functions (DC). Suitable DC decompositions allow us to use the Difference of Convex Algorithm (DCA) in a very efficient way. Our algorithmic approach is used to visualize two real-world datasets.


Data visualization DC functions DC algorithm Multidimensional scaling analysis 

Mathematics Subject Classification

90C90 90C26 



We thank the reviewers for their helpful suggestions and comments, which have been very valuable to strengthen the paper and to improve its quality.


  1. 1.
    Abdi, H., Williams, L.J., Valentin, D., Bennani-Dosse, M.: STATIS and DISTATIS: optimum multitable principal component analysis and three way metric multidimensional scaling. Wiley Interdiscip. Rev. Comput. Stat. 4(2), 124–167 (2012)CrossRefGoogle Scholar
  2. 2.
    Blanquero, R., Carrizosa, E.: Continuous location problems and big triangle small triangle: constructing better bounds. J. Glob. Optim. 45(3), 389–402 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blanquero, R., Carrizosa, E., Hansen, P.: Locating objects in the plane using global optimization techniques. Math. Oper. Res. 34(4), 837–858 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bomze, I.M., Locatelli, M., Tardella, F.: New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. Math. Program. 115(1), 31–64 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications. Springer, Berlin (2005)zbMATHGoogle Scholar
  6. 6.
    Buchin, K., Speckmann, B., Verdonschot, S.: Evolution strategies for optimizing rectangular cartograms. In: Xiao, N., Kwan, M.-P., Goodchild, M.F., Shekhar, S. (eds.) Geographic Information Science, Volume 7478 of Lecture Notes in Computer Science, pp. 29–42. Springer (2012)Google Scholar
  7. 7.
    Cameron, S., Culley, R.: Determining the minimum translational distance between two convex polyhedra. IEEE Int. Conf. Robot. Autom. 3, 591–596 (1986)Google Scholar
  8. 8.
    Carrizosa, E., Conde, E., Muñoz-Márquez, M., Puerto, J.: The generalized Weber problem with expected distances. Revue française d’automatique, d’informatique et de recherche opérationnelle. Recherche opérationnelle 29(1), 35–57 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Carrizosa, E., Dražić, M., Dražić, Z., Mladenović, N.: Gaussian variable neighborhood search for continuous optimization. Comput. Oper. Res. 39(9), 2206–2213 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Carrizosa, E., Guerrero, V.: Biobjective sparse principal component analysis. J. Multivar. Anal. 132, 151–159 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carrizosa, E., Guerrero, V.: rs-Sparse principal component analysis: a mixed integer nonlinear programming approach with VNS. Comput. Oper. Res. 52, 349–354 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Carrizosa, E., Guerrero, V., Romero Morales, D.: A multi-objective approach to visualize adjacencies in weighted graphs by rectangular maps. Technical report, Optimization Online (2015).
  13. 13.
    Carrizosa, E., Guerrero, V., Romero Morales, D.: Visualizing proportions and dissimilarities by space-filling maps: a large neighborhood search approach. Comput. Oper. Res. 78, 369–380 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Carrizosa, E., Martín-Barragán, B., Plastria, F., Romero Morales, D.: On the selection of the globally optimal prototype subset for nearest-neighbor classification. INFORMS J. Comput. 19(3), 470–479 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Carrizosa, E., Muñoz-Márquez, M., Puerto, J.: Location and shape of a rectangular facility in \({\mathbb{R}}^n\). Convexity properties. Math. Program. 83(1–3), 277–290 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Carrizosa, E., Muñoz-Márquez, M., Puerto, J.: The weber problem with regional demand. Eur. J. Oper. Res. 104(2), 358–365 (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Carrizosa, E., Romero Morales, D.: Supervised classification and mathematical optimization. Comput. Oper. Res. 40(1), 150–165 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chen, C.P., Zhang, C.-Y.: Data-intensive applications, challenges, techniques and technologies: a survey on big data. Inf. Sci. 275, 314–347 (2014)CrossRefGoogle Scholar
  19. 19.
    Choo, J., Park, H.: Customizing computational methods for visual analytics with big data. IEEE Comput. Gr. Appl. 33(4), 22–28 (2013)CrossRefGoogle Scholar
  20. 20.
    Cox, T.F., Cox, M.A.A.: Multidimensional Scaling. CRC Press, Boca Raton (2000)zbMATHGoogle Scholar
  21. 21.
    De Leeuw, J., Heiser, W.J.: Convergence of correction matrix algorithms for multidimensional scaling. In: Lingoes, J.C., Roskam, E.E., Borg, I. (eds.) Geometric Representations of Relational Data, pp. 735–752. Mathesis Press, Ann Arbor (1977)Google Scholar
  22. 22.
    De Silva, V., Tenenbaum, J.B.: Sparse Multidimensional Scaling Using Landmark Points. Technical report, Stanford University (2004)Google Scholar
  23. 23.
    Díaz-Báñez, J.M., Mesa, J.A., Schöbel, A.: Continuous location of dimensional structures. Eur. J. Oper. Res. 152(1), 22–44 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Dörk, M., Carpendale, S., Williamson, C.: Visualizing explicit and implicit relations of complex information spaces. Inf. Vis. 11(1), 5–21 (2012)CrossRefGoogle Scholar
  25. 25.
    Dorling, D.: Area cartograms: their use and creation. Concepts and Techniques in Modern Geography Series No. 59. University of East Anglia: Environmental Publications, UK (1996)Google Scholar
  26. 26.
    Ehrgott, M.: A discussion of scalarization techniques for multiple objective integer programming. Ann. Oper. Res. 147(1), 343–360 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Elkeran, A.: A new approach for sheet nesting problem using guided cuckoo search and pairwise clustering. Eur. J. Oper. Res. 231(3), 757–769 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ferrer, A., Martínez-Legaz, J.E.: Improving the efficiency of DC global optimization methods by improving the DC representation of the objective function. J. Glob. Optim. 43(4), 513–531 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Flavin, T., Hurley, M., Rousseau, F.: Explaining stock market correlation: a gravity model approach. Manch. Sch. 70, 87–106 (2002)CrossRefGoogle Scholar
  30. 30.
    Fountoulakis, K., Gondzio, J.: Performance of first- and second-order methods for \(\ell _1\)-regularized least squares problems. Comput. Optim. Appl. 65(3), 605–635 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Fountoulakis, K., Gondzio, J.: A second-order method for strongly convex \(\ell _1\)-regularization problems. Math. Program. 156(1), 189–219 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Gomez-Nieto, E., San Roman, F., Pagliosa, P., Casaca, W., Helou, E.S., de Oliveira, M.C.F., Nonato, L.G.: Similarity preserving snippet-based visualization of web search results. IEEE Trans. Vis. Comput. Gr. 20(3), 457–470 (2014)CrossRefGoogle Scholar
  33. 33.
    Gower, J.C.: Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53(3–4), 325–338 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hansen, P., Jaumard, B.: Cluster analysis and mathematical programming. Math. Program. 79(1–3), 191–215 (1997)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Heilmann, R., Keim, D.A., Panse, C., Sips, M.: Recmap: Rectangular map approximations. In: Proceedings of the IEEE Symposium on Information Visualization, pp. 33–40. IEEE Computer Society (2004)Google Scholar
  36. 36.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)zbMATHGoogle Scholar
  37. 37.
    Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York (1990)CrossRefzbMATHGoogle Scholar
  38. 38.
    Koshizuka, T., Kurita, O.: Approximate formulas of average distances associated with regions and their applications to location problems. Math. Program. 52(1–3), 99–123 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Kruskal, J.B.: Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29(1), 1–27 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Le Thi, H.A., Pham Dinh, T.: D.C. programming approach to the multidimensional scaling problem. In: Migdalas, A., Pardalos, P.M., Värbrand, P. (eds.) From Local to Global Optimization, Volume 53 of Nonconvex Optimizations and Its Applications, pp. 231–276. Springer, Berlin (2001)Google Scholar
  41. 41.
    Le Thi, H.A., Pham Dinh, T.: DC programming approaches for distance geometry problems. In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.) Distance Geometry, pp. 225–290. Springer, Berlin (2013)CrossRefGoogle Scholar
  42. 42.
    Le Thi, H.A.: An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints. Math. Program. 87, 401–426 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Le Thi, H.A., Pham Dinh, T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133(1–4), 23–46 (2005)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Lin, M.C., Manocha, D.: Collision and proximity queries. In: O’Rourke, J., Goodman, E. (eds.) Handbook of Discrete and Computational Geometry. CRC Press, Boca Rotan (2004)Google Scholar
  46. 46.
    Liu, S., Cui, W., Wu, Y., Liu, M.: A survey on information visualization: recent advances and challenges. Vis. Comput. 30(12), 1373–1393 (2014)CrossRefGoogle Scholar
  47. 47.
    Mladenović, N., Dražić, M., Kovačevic-Vujčić, V., Čangalović, M.: General variable neighborhood search for the continuous optimization. Eur. J. Oper. Res. 191(3), 753–770 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Olafsson, S., Li, X., Wu, S.: Operations research and data mining. Eur. J. Oper. Res. 187(3), 1429–1448 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Ong, C.J., Gilbert, E.G.: Growth distances: new measures for object separation and penetration. IEEE Trans. Robot. Autom. 12(6), 888–903 (1996)CrossRefGoogle Scholar
  50. 50.
    Pearson, K.: On lines and planes of closest fit to systems of points in space. Philos. Mag. 2, 559–572 (1901)CrossRefzbMATHGoogle Scholar
  51. 51.
    Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to D.C. programming: theory, algorithms and applications. Acta Math. Vietnam. 22(1), 289–355 (1997)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Pham Dinh, T., Le Thi, H.A.: A branch-and-bound method via DC optimization algorithm and ellipsoidal technique for box constrained nonconvex quadratic programming problems. J. Glob. Optim. 13, 171–206 (1998)CrossRefzbMATHGoogle Scholar
  53. 53.
    Pong, T.K., Tseng, P.: (Robust) edge-based semidefinite programming relaxation of sensor network localization. Math. Program. 130(2), 321–358 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Rabello, R.L., Mauri, G.R., Ribeiro, G.M., Lorena, L.A.N.: A clustering search metaheuristic for the point-feature cartographic label placement problem. Eur. J. Oper. Res. 234(3), 802–808 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    So, A.M.-C., Ye, Y.: Theory of semidefinite programming for sensor network localization. Math. Program. 109(2–3), 367–384 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Speckmann, B., van Kreveld, M., Florisson, S.: A linear programming approach to rectangular cartograms. In: Proceedings of the 12th International Symposium on Spatial Data Handling, pp. 527–546. Springer (2006)Google Scholar
  57. 57.
    Thomas, J., Wong, P.C.: Visual analytics. IEEE Comput. Gr. Appl. 24(5), 20–21 (2004)CrossRefGoogle Scholar
  58. 58.
    Tobler, W.: Thirty five years of computer cartograms. Ann. Assoc. Am. Geogr. 94(1), 58–73 (2004)CrossRefGoogle Scholar
  59. 59.
    Torgerson, W.S.: Theory and Methods of Scaling. Wiley, New York (1958)Google Scholar
  60. 60.
    Trosset, M.W.: Extensions of classical multidimensional scaling via variable reduction. Comput. Stat. 17, 147–163 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Tseng, P.: Second-order cone programming relaxation of sensor network localization. SIAM J. Optim. 18(1), 156–185 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Tuy, H.: Convex Analysis and Global Optimization. Kluwer Academic Publishers, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  63. 63.
    Umetani, S., Yagiura, M., Imahori, S., Imamichi, T., Nonobe, K., Ibaraki, T.: Solving the irregular strip packing problem via guided local search for overlap minimization. Int. Trans. Oper. Res. 16(6), 661–683 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Vaughan, R.: Approximate formulas for average distances associated with zones. Transp. Sci. 18(3), 231–244 (1984)CrossRefGoogle Scholar
  65. 65.
    Wang, Z., Zheng, S., Ye, Y., Boyd, S.: Further relaxations of the semidefinite programming approach to sensor network localization. SIAM J. Optim. 19(2), 655–673 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.IMUS - Instituto de Matemáticas de la Universidad de SevillaSevillaSpain
  2. 2.Copenhagen Business SchoolFrederiksbergDenmark

Personalised recommendations