Skip to main content
SpringerLink
Log in

Enhancing numerical constraint propagation using multiple inclusion representations

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Building tight and conservative enclosures of the solution set is of crucial importance in the design of efficient complete solvers for numerical constraint satisfaction problems (NCSPs). This paper proposes a novel generic algorithm enabling the cooperative use, during constraint propagation, of multiple enclosure techniques. The new algorithm brings into the constraint propagation framework the strength of techniques coming from different areas such as interval arithmetic, affine arithmetic, and mathematical programming. It is based on the directed acyclic graph (DAG) representation of NCSPs whose flexibility and expressiveness facilitates the design of fine-grained combination strategies for general factorable systems. The paper presents several possible combination strategies for creating practical instances of the generic algorithm. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic, and linear programming illustrate the flexibility and efficiency of the approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic, New York (1983)

    MATH  Google Scholar 

  2. Apt, R.K.: The essence of constraint propagation. Theor. Comp. Sci. 221(1–2), 179–210 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Benhamou, F.: Heterogeneous constraint solving. In: Proceedings of 5th International Conference on Algebraic and Logic Programming (ALP’96). LNCS, vol. 1139, pp. 62–76, Aachen, 25–27 September 1996

  4. Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising hull and box consistency. In: Proceedings of the International Conference on Logic Programming (ICLP’99), pp. 230–244, Las Cruces, 29 November–4 December 1999

  5. Benhamou, F., McAllester, D., Van Hentenryck, P.: CLP(Intervals) revisited. In: Proceedings of the International Logic Programming Symposium, pp. 109–123. MIT, Cambridge (1994)

    Google Scholar 

  6. Benhamou, F., Older, W.J.: Applying interval arithmetic to real, integer and Boolean constraints. Technical Report BNR Technical Report, Bell Northern Research, Ontario, Canada (1992)

  7. Benhamou, F., Older, W.J.: Applying interval arithmetic to real, integer and Boolean constraints. J. Log. Program. 32, 1–24 (1997) (Extension of a technical report of Bell Northern Research, Canada, 1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Borradaile, G., Van Hentenryck, P.: Safe and tight linear estimators for global optimization. Math. Program. 42, 2076–2097 (2004)

    Google Scholar 

  9. Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: Proceedings of SIBGRAPI’93, Recife, October 1993

  10. Garloff, J., Jansson, C., Smith, A.P.: Lower bound functions for polynomials. J. Comput. Appl. Math. 157(1), 207–225 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Granvilliers, L., Benhamou, F.: Algorithm 852: RealPaver: an interval solver using constraint satisfaction techniques. ACM Trans. Math. Softw. (TOMS) 32(1), 138–156 (2006)

    Article  MathSciNet  Google Scholar 

  12. Hansen, E.R.: A generalized interval arithmetic. In: Interval Mathematics. LNCS vol. 29, pp. 7–18. Springer, New York (1975)

    Google Scholar 

  13. Hansen, E.R., Walster, G.W.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New York (2004)

    MATH  Google Scholar 

  14. Hongthong, S., Kearfott, R.B.: Rigorous linear overestimators and underestimators. Math. Program. B (2009, in press)

  15. Jansson, C.: Convex-concave extensions. BIT Numer. Math. 40(2), 291–313 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis, 1st edn. Springer, New York (2001)

    MATH  Google Scholar 

  17. Kolev, L.V.: A new method for global solution of systems of non-linear equations. Reliab. Comput. 4, 125–146 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kolev, L.V.: Automatic computation of a linear interval enclosure. Reliab. Comput. 7, 17–18 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kolev, L.V.: An improved interval linearization for solving non-linear problems. In: 10th GAMM – IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN2002), France, September 2002

  20. Lebbah, Y.: ICOS (Interval Constraints Solver). WWW document (2003)

  21. Lebbah, Y., Michel, C., Rueher, M.: Global filtering algorithms based on linear relaxations. In: Notes of the 2nd International Workshop on Global Constrained Optimization and Constraint Satisfaction (COCOS 2003), Switzerland, November 2003

  22. Lebbah, Y., Rueher, M., Michel, C.: A global filtering algorithm for handling systems of quadratic equations and inequations. In: Proceedings of the 9th International Conference on Principles and Practice of Constraint Programming (CP 2003). LNCS, vol. 2470, pp. 109–123. Springer, New York (2003)

    Google Scholar 

  23. Lhomme, O.: Consistency techniques for numeric CSPs. In: Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), pp. 232–238 (1993)

  24. Mackworth, A.K.: Consistency in networks of relations. Artif. Intell. 8, 99–118 (1977)

    Article  MATH  Google Scholar 

  25. Martin, R., Shou, H., Voiculescu, I., Bowyer, A., Wang, G.: Comparison of interval methods for plotting algebraic curves. Comput. Aided Geom. Des. 19(7), 553–587 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  27. McCormick, G.P.: Nonlinear Programming: Theory, Algorithms and Applications. Wiley, New York (1983)

    MATH  Google Scholar 

  28. Messine, F.: New affine forms in interval branch and bound algorithms. Technical Report R2I 99-02, Université de Pau et des Pays de l’Adour (UPPA), France, October 1999

  29. Messine, F.: Extensions of affine arithmetic in interval global optimization algorithms. In: SCAN 2000 and INTERVAL 2000—IMACS/GAMM International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, Germany, 2000

  30. Messine, F.: Extensions of affine arithmetic: Application to unconstrained global optimization. J. Univers. Comput. Sci. 8(11), 992–1015 (2002)

    MathSciNet  Google Scholar 

  31. Miyajima, S., Miyata, T., Kashiwagi, M.: A new dividing method in affine arithmetic. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A(9), 2192–2196 (2003)

    Google Scholar 

  32. Miyajima, S.: On the improvement of the division of the affine arithmetic. Bachelor thesis, Kashiwagi Laboratory, Waseda University, Japan (2000) (It is in Japanese, but easy to guess)

  33. Moore, R.E.: Interval Analysis. Prentice Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  34. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1979)

    MATH  Google Scholar 

  35. Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 2004, 271–369 (2004)

    MathSciNet  Google Scholar 

  36. Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer programming. Math. Program. 99, 283–296 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  38. Schichl, H., Neumaier, A.: Interval analysis on directed acyclic graphs for global optimization. J. Glob. Optim. 33(4), 541–562 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  39. Schichl, H.: Mathematical modeling and global optimization. Habilitation thesis, Faculty of Mathematics, University of Vienna, Autralia, November 2003

  40. Stolfi, J., de Figueiredo, L.H.: Self-validated numerical methods and applications. In: Monograph for 21st Brazilian Mathematics Colloquium (IMPA), Brazil, July 1997

  41. Tawarmalani, M., Sahinidis, N.V.: Convexification and global optimization in continuous and mixed-integer nonlinear programming. Nonconvex Optimization and Its Applications. Kluwer, Deventer (2002)

    Google Scholar 

  42. Van Hentenryck, P.: Numerica: a modeling language for global optimization. In: Proceedings of the 15th International Joint Conference on Artificial Intelligence (IJCAI-97) (1997)

  43. Vu, X.-H., Sam-Haroud, D., Faltings, B.: Combining multiple inclusion representations in numerical constraint propagation. In: Proceedings of the 16th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2004), pp. 458–467. IEEE Computer Society Press, Florida (2004)

    Google Scholar 

  44. Vu, X.-H., Sam-Haroud, D., Silaghi, M.-C.: Numerical constraint satisfaction problems with non-isolated solutions. LNCS, vol. 2861, pp. 194–210. Valbonne-Sophia Antipolis, France, Springer, New York (2003)

  45. Vu, X.-H., Schichl, H., Sam-Haroud, D.: Interval propagation and search on directed acyclic graphs for numerical constraint solving. J. Glob. Optim. (2009, in press)

Download references

Author information

Authors and Affiliations

  1. Cork Constraint Computation Centre, University College Cork, 14 Washington Street West, Cork, Ireland

    Xuan-Ha Vu

  2. Artificial Intelligence Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Batiment IN, Station 14, 1015, Lausanne, Switzerland

    Djamila Sam-Haroud & Boi Faltings

Authors
  1. Xuan-Ha Vu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Djamila Sam-Haroud
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Boi Faltings
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Djamila Sam-Haroud.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vu, XH., Sam-Haroud, D. & Faltings, B. Enhancing numerical constraint propagation using multiple inclusion representations. Ann Math Artif Intell 55, 295 (2009). https://doi.org/10.1007/s10472-009-9129-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10472-009-9129-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation