, Volume 15, Issue 1, pp 93–116 | Cite as

Improving inter-block backtracking with interval Newton

  • Bertrand Neveu
  • Gilles TrombettoniEmail author
  • Gilles Chabert


Inter-block backtracking (IBB) computes all the solutions of sparse systems of nonlinear equations over the reals. This algorithm, introduced by Bliek et al. (1998) handles a system of equations previously decomposed into a set of (small) k ×k sub-systems, called blocks. Partial solutions are computed in the different blocks in a certain order and combined together to obtain the set of global solutions. When solutions inside blocks are computed with interval-based techniques, IBB can be viewed as a new interval-based algorithm for solving decomposed systems of nonlinear equations. Previous implementations used Ilog Solver and its IlcInterval library as a black box, which implied several strong limitations. New versions come from the integration of IBB with the interval-based library Ibex. IBB is now reliable (no solution is lost) while still gaining at least one order of magnitude w.r.t. solving the entire system. On a sample of benchmarks, we have compared several variants of IBB that differ in the way the contraction/filtering is performed inside blocks and is shared between blocks. We have observed that the use of interval Newton inside blocks has the most positive impact on the robustness and performance of IBB. This modifies the influence of other features, such as intelligent backtracking. Also, an incremental variant of inter-block filtering makes this feature more often fruitful.


Intervals Decomposition Solving sparse systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benhamou, F., Goualard, F., Granvilliers, L., & Puget, J.-F. (1999). Revising hull and box consistency. In ICLP (pp. 230–244).Google Scholar
  2. 2.
    Bliek, C., Neveu, B., & Trombettoni, G. (1998). Using graph decomposition for solving continuous CSPs. In Proc. CP’98, LNCS (Vol. 1520, pp. 102–116).Google Scholar
  3. 3.
    Bouma, W., Fudos, I., Hoffmann, C. M., Cai, J., & Paige, R. (1995). Geometric constraint solver. Computer Aided Design, 27(6), 487–501.zbMATHCrossRefGoogle Scholar
  4. 4.
    Chabert, G. (2009). Ibex—An Interval based EXplorer.
  5. 5.
    Chabert, G., & Jaulin, L. (2009). Contractor programming. Artificial Intelligence. Accessed 18 March 2009.Google Scholar
  6. 6.
    Debruyne, R., & Bessière, C. (1997). Some practicable filtering techniques for the constraint satisfaction problem. In Proc. of IJCAI (pp. 412–417).Google Scholar
  7. 7.
    Dechter, R. (1990). Enhancement schemes for constraint processing: Backjumping, learning, and cutset decomposition. Artificial Intelligence, 41(3), 273–312.CrossRefGoogle Scholar
  8. 8.
    Granvilliers, L. (2003). RealPaver user’s manual, version 0.3. University of Nantes.
  9. 9.
    Granvilliers, L., & Benhamou, F. (2006). RealPaver: An interval solver using constraint satisfaction techniques. ACM Transactions on Mathematical Software, 32(1), 138–156.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Hoffmann, C., Lomonossov, A., & Sitharam, M. (1997). Finding solvable subsets of constraint graphs. In Proc. constraint programming CP’97 (pp. 463–477).Google Scholar
  11. 11.
    ILOG, Av. Galliéni, Gentilly (2000). Ilog solver V. 5, users’ reference manual.Google Scholar
  12. 12.
    Jaulin, L., Kieffer, M., Didrit, O., & Walter, E. (2001). Applied interval analysis. New York: Springer.zbMATHGoogle Scholar
  13. 13.
    Jermann, C., Neveu, B., & Trombettoni, G. (2003). Algorithms for identifying rigid subsystems in geometric constraint systems. In Proc. IJCAI (pp. 233–38).Google Scholar
  14. 14.
    Jermann, C., Neveu, B., & Trombettoni, G. (2003). Inter-Block backtracking: Exploiting the structure in continuous CSPs. In Proc. of 2nd int. workshop on global constrained optimization and constraint satisfaction (COCOS’03).Google Scholar
  15. 15.
    Jermann, C., Trombettoni, G., Neveu, B., & Mathis, P. (2006). Decomposition of geometric constraint systems: A survey. International Journal of Computational Geometry and Applications (IJCGA), 16(5–6), 379–414.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Latham, R. S., & Middleditch, A. E. (1996). Connectivity analysis: A tool for processing geometric constraints. Computer Aided Design, 28(11), 917–928.CrossRefGoogle Scholar
  17. 17.
    Lebbah, Y. (1999). Contribution à la résolution de contraintes par consistance forte. Ph.D. thesis, Université de Nantes.Google Scholar
  18. 18.
    Lebbah, Y., Michel, C., Rueher, M., Daney, D., & Merlet, J. P. (2005) Efficient and safe global constraints for handling numerical constraint systems. SIAM Journal on Numerical Analysis, 42(5), 2076–2097.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lhomme, O. (1993). Consistency techniques for numeric CSPs. In IJCAI (pp. 232–238).Google Scholar
  20. 20.
    McAllester, D. A. (1993). Partial order backtracking. Research note, artificial intelligence laboratory, MIT.
  21. 21.
    Merlet, J.-P. (2002). Optimal design for the micro parallel robot MIPS. In Proc. of IEEE international conference on robotics and automation, ICRA ’02, Washington DC, USA (Vol. 2, pp.1149–1154).Google Scholar
  22. 22.
    Neumaier, A. (1990). Interval methods for systems of equations. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  23. 23.
    Neveu, B., Jermann, C., & Trombettoni, G. (2005). Inter-Block backtracking: Exploiting the structure in continuous CSPs. In Selected papers in the 2nd int. worksh. on global constrained optimization and constraints, COCOS, LNCS (Vol. 3478, pp. 15–30).Google Scholar
  24. 24.
    Trombettoni, G., & Chabert, G., (2007). Constructive interval disjunction. In Proc. of CP (pp. 635–650).Google Scholar
  25. 25.
    Van Hentenryck, P., Michel, L., & Deville Y. (1997). Numerica: A modeling language for global optimization. Cambridge: MIT.Google Scholar
  26. 26.
    Wilczkowiak, M., Trombettoni, G., Jermann, C., Sturm, P., & Boyer, E. (2003). Scene modeling based on constraint system decomposition techniques. In Proc. ICCV.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Bertrand Neveu
    • 1
  • Gilles Trombettoni
    • 2
    Email author
  • Gilles Chabert
    • 3
  1. 1.INRIA CERTISSophia Antipolis CedexFrance
  2. 2.INRIA Université de Nice-SophiaSophia Antipolis CedexFrance
  3. 3.LINA Ecole des Mines de NantesSophia Antipolis CedexFrance

Personalised recommendations