Advertisement

Enhancing interval constraint propagation by identifying and filtering n-ary subsystems

  • Ignacio ArayaEmail author
  • Victor Reyes
Article
  • 29 Downloads

Abstract

When interval branch and bound solvers are used for solving numerical constraint satisfaction problems, constraint propagation algorithms are commonly used for filtering/contracting the variable domains. However, these algorithms suffer from the locality problem which is related to the reduced scope of local consistencies. In this work we propose a preprocessing and a filtering technique to reduce the locality problem and to enhance the contraction power of constraint propagation algorithms. The preprocessing consists in constructing a directed acyclic graph (DAG) by merging equivalent nodes (or common subexpressions) and identifying subsystems of n-ary sums in the DAG. The filtering technique consists in applying iteratively HC4 and an ad-hoc technique for contracting the subsystems until reaching a fixed point. Experiments show that the new approach outperforms state-of-the-art strategies using a well known set of benchmark instances.

Keywords

Interval-based solver Common subexpression elimination Constraint propagation Systems of nonlinear equations 

Notes

Acknowledgements

This work is supported by the Fondecyt Project 1160224. Victor Reyes is supported by the Grant Postgrado PUCV 2017.

References

  1. 1.
    Araya, I., Reyes, V.: Interval branch-and-bound algorithms for optimization and constraint satisfaction: a survey and prospects. J. Glob. Optim. 65(4), 837–866 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Mackworth, A.K.: Consistency in networks of relations. Artif. Intell. 8(1), 99–118 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising hull and box consistency. In: International Conference on Logic Programming, Citeseer (1999)Google Scholar
  4. 4.
    Moore, R.: Interval analysis. Series in automatic computation. Prentice-Hall, Englewood Cliff, N.J. (1966)Google Scholar
  5. 5.
    Hansen, E.: Global optimization using interval analysis—the multi-dimensional case. Numer. Math. 34(3), 247–270 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Araya, I., Neveu, B., Trombettoni, G.: Exploiting common subexpressions in numerical CSPs. In: Principles and Practice of Constraint Programming (CP 2008). Springer, pp. 342–357 (2008)Google Scholar
  7. 7.
    Schichl, H., Neumaier, A.: Interval analysis on directed acyclic graphs for global optimization. J. Glob. Optim. 33(4), 541–562 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ceberio, M., Granvilliers, L.: Solving nonlinear equations by abstraction, Gaussian elimination, and interval methods. In: Frontiers of Combining Systems. Springer, pp. 117–131 (2002)Google Scholar
  9. 9.
    Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer linear programming. Math. Program. 99(2), 283–296 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Baharev, A., Achterberg, T., Rév, E.: Computation of an extractive distillation column with affine arithmetic. AIChE J. 55(7), 1695–1704 (2009)CrossRefGoogle Scholar
  11. 11.
    Chabert, G., Jaulin, L.: Contractor programming. Artif. Intell. 173, 1079–1100 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Neveu, B., Trombettoni, G., et al.: Adaptive constructive interval disjunction. In: International Conference on Tools with Artificial Intelligence (ICTAI), pp. 900–906 (2013)Google Scholar
  13. 13.
    Hansen, E.: Global Optimization Using Interval Analysis. Marcel Dekker, New York (1992)zbMATHGoogle Scholar
  14. 14.
    Trombettoni, G., Araya, I., Neveu, B., Chabert, G.: Inner regions and interval linearizations for global optimization. In: AAAI Conference on Artificial Intelligence, pp. 99–104 (2011)Google Scholar
  15. 15.
    Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. 4OR, vol. 13, no. 3, pp. 247–277 (2015)Google Scholar
  16. 16.
    Araya, I., Trombettoni, G., Neveu, B.: A contractor based on convex interval Taylor. In: Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. Springer, pp. 1–16 (2012)Google Scholar
  17. 17.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Pontificia Universidad Católica de ValparaísoValparaisoChile

Personalised recommendations