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Filtering Domains of Factorable Functions Using Interval Contractors

  • Laurent GranvilliersEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

Many theorems in mathematics require a real function to be continuous over the domain under consideration. In particular the Brouwer fixed point theorem and the mean value theorem underlie many interval methods like Newton operators for solving numerical constraint satisfaction problems or global optimization problems. Since the continuity property collapses when the function is not defined at some point it is then important to check whether the function is defined everywhere in a given domain. We introduce here an interval branch-and-contract algorithm that rigorously approximate the domain of definition of a factorable function within a box. The proposed approach mainly relies on interval contractors applied to the domain constraints and their negations stemming from the expression of the function.

Keywords

Interval methods Branch-and-contract algorithm Interval contractor Constraint satisfaction problem Paving 

Notes

Acknowledgment

The author would like to thank Christophe Jermann for interesting discussions about these topics and his careful reading of a preliminary version of this paper.

References

  1. 1.
    1788-2015, I.S.: IEEE Standard for Interval Arithmetic (2015)Google Scholar
  2. 2.
    Benhamou, F., Goualard, F.: Universally quantified interval constraints. In: Proceedings of International Conference on Principles and Practice of Constraint Programming (CP), pp. 67–82 (2000)CrossRefGoogle Scholar
  3. 3.
    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.F.: Revising hull and box consistency. In: Proceedings of International Conference on Logic Programming (ICLP), pp. 230–244 (1999)Google Scholar
  4. 4.
    Chabert, G., Beldiceanu, N.: Sweeping with continuous domains. In: Proceedings of International Conference on Principles and Practice of Constraint Programming (CP), pp. 137–151 (2010)CrossRefGoogle Scholar
  5. 5.
    Chabert, G., Jaulin, L.: Contractor programming. Artif. Intell. 173(11), 1079–1100 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Collavizza, H., Delobel, F., Rueher, M.: Extending consistent domains of numeric CSP. In: Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), pp. 406–413 (1999)Google Scholar
  7. 7.
    Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33(2) (2007)CrossRefGoogle Scholar
  8. 8.
    Granvilliers, L.: A new interval contractor based on optimality conditions for bound constrained global optimization. In: Proceedings of International Conference on Tools with Artificial Intelligence (ICTAI), pp. 90–97 (2018)Google Scholar
  9. 9.
    Granvilliers, L., Benhamou, F.: Algorithm 852: realpaver: an interval solver using constraint satisfaction techniques. ACM Trans. Math. Softw. 32(1), 138–156 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hentenryck, P.V., McAllester, D., Kapur, D.: Solving polynomial systems using a branch and prune approach. SIAM J. Numer. Anal. 34(2), 797–827 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lhomme, O.: Consistency techniques for numeric CSPs. In: Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), pp. 232–238 (1993)Google Scholar
  12. 12.
    Mackworth, A.K.: Consistency in networks of relations. Artif. Intell. 8, 99–118 (1977)CrossRefGoogle Scholar
  13. 13.
    Moore, R.E.: Interval Analysis. Prentice-Hall (1966)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.LS2N, Université de NantesNantesFrance

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