Advertisement

Constraints

, Volume 20, Issue 4, pp 452–467 | Cite as

Adaptive constructive interval disjunction: algorithms and experiments

  • Bertrand Neveu
  • Gilles Trombettoni
  • Ignacio Araya
Article

Abstract

An operator called CID and an efficient variant 3BCID were proposed in 2007. For the numerical CSP handled by interval methods, these operators compute a partial consistency equivalent to Partition-1-AC for the discrete CSP. In addition to the constraint propagation procedure used to refute a given subproblem, the main two parameters of CID are the number of times the main CID procedure is called and the maximum number of sub-intervals treated by the procedure. The 3BCID operator is state-of-the-art in numerical CSP, but not in constrained global optimization, for which it is generally too costly. This paper proposes an adaptive variant of 3BCID called ACID. The number of variables handled is auto-adapted during the search, the other parameters are fixed and robust to modifications. On a representative sample of instances, ACID appears to work efficiently, both with the HC4 constraint propagation algorithm and with the state-of-the-art Mohc algorithm. Experiments also highlight that it is relevant to auto-adapt only a number of handled variables, instead of a specific set of selected variables. Finally, ACID appears to be the best interval constraint programming operator for solving and optimization, and has been therefore added to the default strategies of the Ibex interval solver.

Keywords

Interval methods Adaptive algorithms Strong consistency 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Araya, I., Trombettoni, G., Neveu, B. (2010). Exploiting monotonicity in interval constraint propagation. In Proceeding of AAAI (pp. 9–14).Google Scholar
  2. 2.
    Araya, I., Trombettoni, G., Neveu, B. (2010). Making adaptive an interval constraint propagation algorithm exploiting monotonicity. In Proceedings of CP, Constraint Programming, LNCS 6308 (pp. 61–68). Springer.Google Scholar
  3. 3.
    Araya, I., Trombettoni, G., Neveu, B. (2012). A Contractor based on convex interval Taylor. In CPAIOR 2012, no. 7298 in LNCS (pp. 1–16).Google Scholar
  4. 4.
    Balafrej, A., Bessiere, C., Bouyakhf, E., Trombettoni, G. (2014). Adaptive singleton-based consistencies. In AAAI (pp. 2601–2607). AAAI Press.Google Scholar
  5. 5.
    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.F. (1999). Revising hull and box consistency. In Proceedings of ICLP (pp. 230–244).Google Scholar
  6. 6.
    Bennaceur, H., & Affane, M.S. (2001). Partition-k-AC: an efficient filtering technique combining domain partition and arc consistency. In Proceedings of CP (pp. 560–564).Google Scholar
  7. 7.
    Bessiere, C., & Debruyne, R. (2005). Optimal and suboptimal singleton arc consistency algorithms. In Proceedings of IJCAI (pp. 54–59).Google Scholar
  8. 8.
    Chabert, G., & Jaulin, L. (2009). Contractor programming. Artificial Intelligence, 173, 1079–1100.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chabert, G., & Jaulin, L. (2009). Hull consistency under monotonicity. In Proceedings of CP, LNCS 5732 (pp. 188–195).Google Scholar
  10. 10.
    Debruyne, R., & Bessiere, C. (1997). Some practicable filtering techniques for the constraint satisfaction problem. In Proceedings of IJCAI (pp. 412–417).Google Scholar
  11. 11.
    Hansen, E. (1992). Global optimization using interval analysis. Marcel Dekker Inc.Google Scholar
  12. 12.
    Kieffer, M., Jaulin, L., Walter, E., Meizel, D. (2000). Robust autonomous robot localization using interval analysis. Reliable Computing, 3(6), 337–361.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lebbah, Y., Michel, C., Rueher, M., Daney, D., Merlet, J. (2005). Efficient and safe global constraints for handling numerical constraint systems. SIAM Journal on Numerical Analysis, 42(5), 2076–2097.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lhomme, O. (1993). Consistency techniques for numeric CSPs. In IJCAI (pp. 232–238).Google Scholar
  15. 15.
    Mackworth, A. (1977). Consistency in networks of relations. Artificial Intelligence, 8, 99–118.CrossRefzbMATHGoogle Scholar
  16. 16.
    Merlet, J.P. (2007). Interval analysis and robotics. In Symposium of Robotics Research.Google Scholar
  17. 17.
    Messine, F. (1997). Méthodes d’optimisation globale basées sur l’analyse d’intervalle pour la résolution des problèmes avec contraintes. Ph.D. thesis, LIMA-IRIT-ENSEEIHT-INPT, Toulouse.Google Scholar
  18. 18.
    Messine, F. (2002). Extensions of affine arithmetic: application to global optimization. Journal of Universal Computer Science, 8(11), 992–1015.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Messine, F., & Laganouelle, J.L. (1998). Enclosure methods for multivariate differentiable functions and application to global optimization. Journal of Universal Computer Science, 4(6), 589–603.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Min Li, C. (1997). Anbulagan: Heuristics based on unit propagation for satisfiability problems. In Proceedings IJCAI (pp. 366–371).Google Scholar
  21. 21.
    Tawarmalani, M., & Sahinidis, N.V. (2005). A polyhedral branch-and-cut approach to global optimization. Mathematical Programming, 103(2), 225–249.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Trombettoni, G., Araya, I., Neveu, B., Chabert, G. (2011). Inner regions and interval linearizations for global optimization. In AAAI (pp. 99–104).Google Scholar
  23. 23.
    Trombettoni, G., & Chabert, G. (2007). Constructive interval disjunction. In Proceedings of CP, LNCS 4741 (pp. 635–650).Google Scholar
  24. 24.
    Tucker, W. (2002). A rigorous ODE solver and smale’s 14th problem. Foundations of Computational Mathematics, 2, 53–117.MathSciNetCrossRefzbMATHGoogle 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 New York 2015

Authors and Affiliations

  • Bertrand Neveu
    • 1
  • Gilles Trombettoni
    • 2
  • Ignacio Araya
    • 3
  1. 1.Imagine LIGM Université Paris–EstParisFrance
  2. 2.LIRMM, University of Montpellier, CNRSMontpellierFrance
  3. 3.Pontificia Universidad Católica de ValparaísoValparaisoChile

Personalised recommendations