Skip to main content
Log in

Validated Constraint Solving—Practicalities, Pitfalls, and New Developments

Reliable Computing

Abstract

Many constraint propagation techniques iterate through the constraints in a straightforward manner, but can fail because they do not take account of the coupling between the constraints.However, some methods of taking account of this coupling are local in nature, and fail if the initial search region is too large.We put into perspective newer methods, based on linear relaxations, that can often replace brute-force search with the solution of a large, sparse linear program.

Robustness has been recognized as important in geometric computations and elsewhere for at least a decade, and more and more developers are including validation in the design of their systems. We provide citations to our work and to the work of others to-date in developing validated versions of linear relaxations.

This work is in the form of a brief review and prospectus for future development. We give various simple examples to illustrate our points.

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

Access this article

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. Babichev, A. B., Kadyrova, O. B., Kashevarova, T. P., Leshchenko, A. S., and Semenov, A. L.: UniCalc, A Novel Approach to Solving Systems of Algebraic Equations, Interval Computations 2 (1993), pp. 29–47.

    Google Scholar 

  2. Benhamou, F.: Interval Constraints, Interval Propagation, in Floudas, C. and Pardalos, P. (eds), Encyclopedia of Optimization, Kluwer Academic Publishers, Dordrecht, 2001.

    Google Scholar 

  3. Berz, M.: Cosy Infinity Web Page, 2000, http://cosy.pa.msu.edu/.

  4. Cleary, J. G.: Logical Arithmetic, Future Computing Systems 2 (2) (1987), pp. 125–149.

    Google Scholar 

  5. Floudas, C. A.: Deterministic Global Optimization: Theory, Algorithms and Applications, Kluwer Academic Publishers, Dordrecht, 2000.

    Google Scholar 

  6. Hongthong, S. and Kearfott, R. B.: Rigorous Linear Overestimators and Underestimators, preprint, 2004, http://interval.louisiana.edu/preprints/estimates_of_powers.pdf.

  7. Jansson, Ch.:A Rigorous Lower Bound for the Optimal Value of Convex Optimization Problems, J. Global Optim. 28 (1) (2004), pp. 121–137.

    Article  Google Scholar 

  8. Kearfott, R. B.: Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for Nonlinear Systems, Computing 47 (2) (1991), pp. 169–191.

    Google Scholar 

  9. Kearfott, R. B.: Empirical Comparisons of Linear Relaxations and Alternate Techniques in Validated Deterministic Global Optimization, preprint, 2004, http://interval.louisiana.edu/preprints/validated_global_optimization_search_comparisons.pdf.

  10. Kearfott, R. B.: Globsol: History, Composition, and Advice on Use, in: Global Optimization and Constraint Satisfaction, Lecture Notes in Computer Science, Springer-Verlag, New York, 2003, pages 17–31.

  11. Kearfott, R. B.: Interval Analysis: Interval Newton Methods, in: Encyclopedia of Optimization, volume 3, Kluwer Academic Publishers, 2001, pp. 76–78.

  12. Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.

    Google Scholar 

  13. Kearfott, R. B. and Hongthong, S.: A Preprocessing Heuristic for Determining the Difficulty of and Selecting a Solution Strategy for Nonconvex Optimization, preprint, 2003, http://interval.louisiana.edu/preprints/2003_symbolic_analysis_of_GO.pdf.

  14. Kearfott, R. B., Neher, M., Oishi, S., and Rico, F.: Libraries, Tools, and Interactive Systems for Verified Computations: Four Case Studies, in: Alt, R., Frommer, A., Kearfott, R. B., and Luther, W. (eds), Numerical Software with Result Verification, Lecture Notes in Computer Science 2991, Springer-Verlag, New York, 2004, pp. 36–63.

  15. Kearfott, R. B. and Walster, G.W.: Symbolic Preconditioning with Taylor Models: Some Examples, Reliable Computing 8 (6) (2002), pp. 453–468.

    Article  Google Scholar 

  16. Kreinovich, V., Lakeyev, A., Rohn, J., and Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer Academic Publishers, Dordrecht, 1998.

    Google Scholar 

  17. Lebbah, Y., Michel, C., Rueher, M., Daney, D., and Merlet, J.-P.: Efficient and Safe Global Constraints for Handling Numerical Constraint Systems, SIAM J. Numer. Anal., accepted for publication.

  18. Narin’yani, A. S.: Intelligent Software Technology for the New Decade, Comm. ACM 34 (6) (1991), pp. 60–67.

    Article  Google Scholar 

  19. Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  20. Neumaier, A. and Shcherbina, O.: Safe Bounds in Linear and Mixed-Integer Programming, Math. Prog. 99 (2) (2004), pp. 283–296, http://www.mat.univie.ac.at/~neum/ms/mip.pdf.

  21. Rump, S. M. et al.: INTLAB home page, 2000, http://www.ti3.tu-harburg.de/~rump/intlab/index.html.

  22. Tawarmalani, M. and Sahinidis, N. V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, Dordrecht, 2002.

    Google Scholar 

  23. Van Hentenryck, P., Michel, L., and Deville, Y.: Numerica: A Modeling Language for Global Optimization, MIT Press, Cambridge, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Baker Kearfott.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kearfott, R.B. Validated Constraint Solving—Practicalities, Pitfalls, and New Developments. Reliable Comput 11, 383–391 (2005). https://doi.org/10.1007/s11155-005-0045-0

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11155-005-0045-0

Keywords

Navigation