Advertisement

Abstract

Interval Taylor has been proposed in the sixties by the interval analysis community for relaxing continuous non-convex constraint systems. However, it generally produces a non-convex relaxation of the solution set. A simple way to build a convex polyhedral relaxation is to select a corner of the studied domain/box as expansion point of the interval Taylor form, instead of the usual midpoint. The idea has been proposed by Neumaier to produce a sharp range of a single function and by Lin and Stadtherr to handle n ×n (square) systems of equations.

This paper presents an interval Newton-like operator, called X-Newton, that iteratively calls this interval convexification based on an endpoint interval Taylor. This general-purpose contractor uses no preconditioning and can handle any system of equality and inequality constraints. It uses Hansen’s variant to compute the interval Taylor form and uses two opposite corners of the domain for every constraint.

The X-Newton operator can be rapidly encoded, and produces good speedups in constrained global optimization and constraint satisfaction. First experiments compare X-Newton with affine arithmetic.

Keywords

Global Optimization Constraint Satisfaction Interval Arithmetic Simplex Algorithm Opposite Corner 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aberth, O.: The Solution of Linear Interval Equations by a Linear Programming Method. Linear Algebra and its Applications 259, 271–279 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Araya, I., Trombettoni, G., Neveu, B.: Exploiting Monotonicity in Interval Constraint Propagation. In: Proc. AAAI, pp. 9–14 (2010)Google Scholar
  3. 3.
    Araya, I., Trombettoni, G., Neveu, B.: A Contractor Based on Convex Interval Taylor. Technical Report 7887, INRIA (February 2012)Google Scholar
  4. 4.
    Baharev, A., Achterberg, T., Rév, E.: Computation of an Extractive Distillition Column with Affine Arithmetic. AIChE Journal 55(7), 1695–1704 (2009)CrossRefGoogle Scholar
  5. 5.
    Beaumont, O.: Algorithmique pour les intervalles. PhD thesis, Université de Rennes (1997)Google Scholar
  6. 6.
    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising Hull and Box Consistency. In: Proc. ICLP, pp. 230–244 (1999)Google Scholar
  7. 7.
    Bliek, C.: Computer Methods for Design Automation. PhD thesis, MIT (1992)Google Scholar
  8. 8.
    Chabert, G.: Techniques d’intervalles pour la résolution de systèmes d’intervalles. PhD thesis, Université de Nice–Sophia (2007)Google Scholar
  9. 9.
    Chabert, G., Jaulin, L.: Contractor Programming. Artificial Intelligence 173, 1079–1100 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    de Figueiredo, L., Stolfi, J.: Affine Arithmetic: Concepts and Applications. Numerical Algorithms 37(1-4), 147–158 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Goldsztejn, A., Granvilliers, L.: A New Framework for Sharp and Efficient Resolution of NCSP with Manifolds of Solutions. Constraints (Springer) 15(2), 190–212 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hansen, E.: Global Optimization using Interval Analysis. Marcel Dekker Inc. (1992)Google Scholar
  13. 13.
    Hansen, E.R.: On Solving Systems of Equations Using Interval Arithmetic. Mathematical Comput. 22, 374–384 (1968)zbMATHCrossRefGoogle Scholar
  14. 14.
    Hansen, E.R.: Bounding the Solution of Interval Linear Equations. SIAM J. Numerical Analysis 29(5), 1493–1503 (1992)zbMATHCrossRefGoogle Scholar
  15. 15.
    Kearfott, R.B.: Rigorous Global Search: Continuous Problems. Kluwer Academic Publishers (1996)Google Scholar
  16. 16.
    Kreinovich, V., Lakeyev, A.V., Rohn, J., Kahl, P.T.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer (1997)Google Scholar
  17. 17.
    Lebbah, Y., Michel, C., Rueher, M.: An Efficient and Safe Framework for Solving Optimization Problems. J. Computing and Applied Mathematics 199, 372–377 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lebbah, Y., Michel, C., Rueher, M., Daney, D., Merlet, J.P.: Efficient and safe global constraints for handling numerical constraint systems. SIAM Journal on Numerical Analysis 42(5), 2076–2097 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lin, Y., Stadtherr, M.: LP Strategy for the Interval-Newton Method in Deterministic Global Optimization. Industrial & Engineering Chemistry Research 43, 3741–3749 (2004)CrossRefGoogle Scholar
  20. 20.
    McAllester, D., Van Hentenryck, P., Kapur, D.: Three Cuts for Accelerated Interval Propagation. Technical Report AI Memo 1542, Massachusetts Institute of Technology (1995)Google Scholar
  21. 21.
    Messine, F., Laganouelle, J.-L.: Enclosure Methods for Multivariate Differentiable Functions and Application to Global Optimization. Journal of Universal Computer Science 4(6), 589–603 (1998)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Moore, R.E.: Interval Analysis. Prentice-Hall (1966)Google Scholar
  23. 23.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM (2009)Google Scholar
  24. 24.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ. Press (1990)Google Scholar
  25. 25.
    Neumaier, A., Shcherbina, O.: Safe Bounds in Linear and Mixed-Integer Programming. Mathematical Programming 99, 283–296 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ninin, J., Messine, F., Hansen, P.: A Reliable Affine Relaxation Method for Global Optimization. research report RT-APO-10-05, IRIT (March 2010) (submitted)Google Scholar
  27. 27.
    Oettli, W.: On the Solution Set of a Linear System with Inaccurate Coefficients. SIAM J. Numerical Analysis 2(1), 115–118 (1965)MathSciNetGoogle Scholar
  28. 28.
    Schaefer, T.J.: The Complexity of Satis ability Problems. In: Proc. STOC, ACM Symposium on Theory of Computing, pp. 216–226 (1978)Google Scholar
  29. 29.
    Tawarmalani, M., Sahinidis, N.V.: A Polyhedral Branch-and-Cut Approach to Global Optimization. Mathematical Programming 103(2), 225–249 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Trombettoni, G., Araya, I., Neveu, B., Chabert, G.: Inner Regions and Interval Linearizations for Global Optimization. In: AAAI, pp. 99–104 (2011)Google Scholar
  31. 31.
    Trombettoni, G., Chabert, G.: Constructive Interval Disjunction. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 635–650. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  32. 32.
    Vu, X.-H., Sam-Haroud, D., Faltings, B.: Enhancing Numerical Constraint Propagation using Multiple Inclusion Representations. Annals of Mathematics and Artificial Intelligence 55(3-4), 295–354 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ignacio Araya
    • 1
  • Gilles Trombettoni
    • 2
  • Bertrand Neveu
    • 3
  1. 1.UTFSMChile
  2. 2.IRIT, INRIA, I3SUniversité Nice-SophiaFrance
  3. 3.Imagine LIGM Université Paris-EstFrance

Personalised recommendations