, Volume 17, Issue 4, pp 432–460 | Cite as

Interval-based projection method for under-constrained numerical systems

  • Daisuke Ishii
  • Alexandre Goldsztejn
  • Christophe Jermann


This paper presents an interval-based method that follows the branch-and-prune scheme to compute a verified paving of a projection of the solution set of an under-constrained system. Benefits of this algorithm include anytime solving process, homogeneous verification of inner boxes, and applicability to generic problems, allowing any number of (possibly nonlinear) equality and inequality constraints. We present three key improvements of the algorithm dedicated to projection problems: (i) The verification process is enhanced in order to prove faster larger boxes in the projection space. (ii) Computational effort is saved by pruning redundant portions of the solution set that would project identically. (iii) A dedicated branching strategy allows reducing the number of treated boxes. Experimental results indicate that various applications can be modeled as projection problems and can be solved efficiently by the proposed method.


Numerical constraint programming Interval analysis Under-constrained systems Projection method Existentially quantified constraints 


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  1. 1.
    Araya, I., Trombettoni, G., Neveu, B. (2010). Exploiting monotonicity in interval constraint propagation. In Proc. of AAAI’10.Google Scholar
  2. 2.
    Beltran, M., Castillo, G., Kreinovich, V. (1998). Algorithms that still produce a solution (maybe not optimal) even when interrupted: Shary’s idea justified. Reliable Computing, 4(1), 39–53.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Benhamou, F., McAllester, D., Van Hentenryck, P. (1994). CLP(intervals) revisited. In Proc. of Intl. Symp. on Logic Prog (pp. 124–138). The MIT Press.Google Scholar
  4. 4.
    Collins, G.E. (1998). Quantifier elimination by cylindrical algebraic decomposition—twenty years of progress. Quantifier Elimination and Cylindrical Algebraic Decomposition (pp. 8–23).Google Scholar
  5. 5.
    Goldsztejn, A. (2006). A branch and prune algorithm for the approximation of non-linear ae-solution sets. In Proc. of ACM SAC 2006 (pp. 1650–1654).Google Scholar
  6. 6.
    Goldsztejn, A., & Jaulin, L. (2006). Inner and outer approximations of existentially quantified equality constraints. In Proc. of CP’06, LNCS4204 (pp. 198–212).Google Scholar
  7. 7.
    Goldsztejn, A., & Jaulin, L. (2010). Inner approximation of the range of vector-valued functions. Reliable Computing, 14, 1–23.MathSciNetGoogle Scholar
  8. 8.
    Goualard, F. (2008). Gaol: NOT just another interval library (version 3.1.1). Accessed 3 Aug 2012
  9. 9.
    Granvilliers, L. (2010). Realpaver (version 1.1). Accessed 3 Aug 2012
  10. 10.
    Hansen, E., & Sengupta, S. (1981) Bounding solutions of systems of equations using interval analysis. BIT, 21, 203–211.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Herrero, P., Jaulin, L., Vehi, J., Sainz, M. (2010). Guaranteed set-point computation with application to the control of a sailboat. International Journal of Control, Automation and Systems, 8(1), 1–7.CrossRefGoogle Scholar
  12. 12.
    Herrero, P., Sainz, M.A., Veh, J., Jaulin, L. (2005). Quantified set inversion algorithm with applications to control. Reliable Computing, 11(5), 369–382.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, E. (2001). Applied interval analysis, with examples in parameter and state estimation, robust control and robotics. Springer.Google Scholar
  14. 14.
    Kearfott, R.B., Nakao, M.T., Neumaier, A., Rump, S.M., Shary, S.P., Van Hentenryck, P. (2005) Standardized notation in interval analysis. In Proc. of XIII Baikal International School-seminar “Optimization methods and their applications” (pp. 106–113).Google Scholar
  15. 15.
    Kearfott, R.B., & Novoa III, M. (1990). Algorithm 681: intbis, a portable interval newton/bisection package. ACM Transactions on Mathematical Software, 16(2), 152–157.zbMATHCrossRefGoogle Scholar
  16. 16.
    Khalil, H.K. (2002). Nonlinear systems, (3rd edn). Prentice Hall.Google Scholar
  17. 17.
    Lhomme, O. (1993). Consistency techniques for numeric CSPs. In Proc. of the 13th International Joint Conference on Artificial Intelligence (IJCAI’93) (pp. 232–238).Google Scholar
  18. 18.
    Moore, R. (1966). Interval analysis. Prentice-Hall.Google Scholar
  19. 19.
    Neumaier, A. (1988). The enclosure of solutions of parameter-dependent systems of equations. In R. Moore (Ed.), Reliability in Computing (pp. 269–286). San Diego: Academic Press.Google Scholar
  20. 20.
    Neumaier, A. (1990). Interval methods for systems of equations. Cambridge University Press.Google Scholar
  21. 21.
    Ratschan, S. (2000). Uncertainty propagation in heterogeneous algebras for approximate quantified constraint solving. Journal of Universal Computer Science 6(9), 861–880.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Reboulet, C. (1988). Modélisation des robots parallèles. In J.D. Boissonat, B. Faverjon, J.P. Merlet (Eds.), Techniques de la robotique, architecture et commande (pp. 257–284). Paris, France: Hermes sciences.Google Scholar
  23. 23.
    Rossi, F., van Beek, P., Walsh, T. (2006). Handbook of constraint programming (Foundations of Artificial Intelligence). New York, NY, USA: Elsevier Science Inc.Google Scholar
  24. 24.
    Shary, S.P. (2002) A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Computing 8(5), 321–418.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Van Hentenryck, P., Michel, L., Deville, Y. (1997). Numerica: A modeling language for global optimization. MIT Press.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Daisuke Ishii
    • 1
  • Alexandre Goldsztejn
    • 2
  • Christophe Jermann
    • 3
  1. 1.JSPS, National Institute of InformaticsTokyoJapan
  2. 2.CNRS, LINA (UMR 6241)NantesFrance
  3. 3.University of Nantes, LINA (UMR 6241)NantesFrance

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