Journal of Global Optimization

, Volume 66, Issue 1, pp 111–126 | Cite as

Maximum consistency method for data fitting under interval uncertainty



The work is devoted to application of global optimization in data fitting problem under interval uncertainty. Parameters of the linear function that best fits intervally defined data are taken as the maximum point for a special (“recognizing”) functional which is shown to characterize consistency between the data and parameters. The new data fitting technique is therefore called “maximum consistency method”. We investigate properties of the recognizing functional and present interpretation of the parameter estimates produced by the maximum consistency method.


Data fitting Interval uncertainty Recognizing functional Maximum consistency Nonconvex optimization 

Mathematics Subject Classification

62H99 93B30 90C26 


  1. 1.
    Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)MATHGoogle Scholar
  2. 2.
    Beeck, H.: Über die Struktur und Abschätzungen der Lösungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten. Computing 10, 231–244 (1972)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Draper, N.R., Smith, H.: Applied Regression Analysis, 3rd edn. Wiley, New York (1998)MATHGoogle Scholar
  4. 4.
    Fiedler, M., Nedoma, J., Ramik, J., Rohn, J., Zimmerman, M.: Linear Optimization Problems with Inexact Data. Springer, Berlin (2006)MATHGoogle Scholar
  5. 5.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis, with Examples in Parameter and State Estimation. Robust Control and Robotics. Springer, London (2001)MATHGoogle Scholar
  6. 6.
    Kantorovich, L. V.: On some new approaches to numerical methods and processing observation data. Sib. Math. J. 3(5), 701–709 (1962) (in Russian). Electronic version is accessible at
  7. 7.
    Kearfott, R.B., Nakao, M., Neumaier, A., Rump, S., Shary, S.P., van Hentenryck, P.: Standardized notation in interval analysis. Comput. Technol. 15(1), 7–13 (2010)MATHGoogle Scholar
  8. 8.
    Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1996)MATHGoogle Scholar
  9. 9.
    Lakeyev, A.V., Noskov, S.I.: A description of the set of solutions of a linear equation with intervally defined operator and right-hand side. Russian Acad. Sci. Dokl. Math. 47(3), 518–523 (1993)MathSciNetGoogle Scholar
  10. 10.
    Milanese, M., Norton, J., Piet-Lahanier, H., Walter, E. (eds.): Bounding Approaches to System Identification. Plenum Press, New York (1996)MATHGoogle Scholar
  11. 11.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)CrossRefMATHGoogle Scholar
  12. 12.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  13. 13.
    Nurminski, E.A.: Separating plane algorithms for convex optimization. Math. Progr. 76(3), 373–391 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rao, C.R., Toutenburg, H., Shalabh, Heumann, C.: Linear Models and Generalizations. Least Squares and Alternatives. Springer, New York (2008)MATHGoogle Scholar
  15. 15.
    Remez, E.Ya.: General computational methods of Chebyshev approximation: The problems with linear real parameters. Oak Ridge, U.S. Atomic Energy Commission. Translation 4491 (1962)Google Scholar
  16. 16.
    Schweppe, F.C.: Recursive state estimation: unknown but bounded errors and system inputs. IEEE Trans. Autom. Control 13(1), 22–28 (1968)CrossRefGoogle Scholar
  17. 17.
    Shary, S.P.: Solvability of interval linear equations and data analysis under uncertainty. Autom. Remote Control 73(2), 310–322 (2012). doi: 10.1134/S0005117912020099 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Shary, S.P.: Finite-dimensional interval analysis. Institute of Computational Technologies SB RAS, Novosibirsk (2013). Electronic book accessible at
  19. 19.
    Shary, S.P., Sharaya, I.A.: Recognition of solvability of interval equations and its application to data analysis. Comput. Technol. 18(3), 80–109 (2013) (in Russian)Google Scholar
  20. 20.
    Shary, S.P., Sharaya, I.A.: On solvability recognition for interval linear systems of equations. Optim. Lett. (2015). Prepublished on May 6, 2015. doi:  10.1007/s11590-015-0891-6
  21. 21.
    Shor, N.Z.: Nondifferentiable Optimization and Polynomial Problems. Kluwer, Boston (1998)CrossRefMATHGoogle Scholar
  22. 22.
    Shor, N.Z., Stetsyuk, P.I.: Modified \(r\)-algorithm to find the global minimum of polynomial functions. Cybern. Syst. Anal. 33(4), 482–497 (1997)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Strekalovsky, A.S.: On the minimization of the difference of convex functions on a feasible set. Comput. Math. Math. Physics 43(3), 380–390 (2003)MathSciNetGoogle Scholar
  24. 24.
    Vorontsova, E.A.: A projective separating plane method with additional clipping for non-smooth optimization. WSEAS Trans. Math. 13, 115–121 (2014)Google Scholar
  25. 25.
    Zhilin, S.I.: On fitting empirical data under interval error. Reliab. Comput. 11(5), 433–442 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of Computational TechnologiesNovosibirskRussia

Personalised recommendations