Skip to main content
SpringerLink
Log in

Rigorous Global Filtering Methods with Interval Unions

  • Chapter
  • First Online:
Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications

Part of the book series: Studies in Computational Intelligence ((SCI,volume 835))

Abstract

This paper presents rigorous filtering methods for constraint satisfaction problems based on the interval union arithmetic. Interval unions are finite sets of closed and disjoint intervals that generalize the interval arithmetic. They allow a natural representation of the solution set of interval powers, trigonometric functions and the division by intervals containing zero. We show that interval unions are useful when applied to the forward-backward constraint propagation on directed acyclic graphs (DAGs) and can also replace the interval arithmetic in the Newton operator. Empirical observations support the conclusion that interval unions reduce the search domain even when more expensive state-of-the-art methods fail. Interval unions methods tend to produce a large number of boxes at each iteration. We address this problem by taking a suitable gap-filling strategy. Numerical experiments on constraint satisfaction problems from the COCONUT show the capabilities of the new approach.

This research was supported through the research grants P25648-N25 of the Austrian Science Fund (FWF) and 853930 of the Austrian Research Promotion Agency (FFG). Dedicated to Vladik Kreinovich on the occasion of his 65th birthday.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. F. Domes, GloptLab - a configurable framework for the rigorous global solution of quadratic constraint satisfaction problems. Optim. Methods Softw. 24, 727–747 (2009)

    Article  MathSciNet  Google Scholar 

  2. F. Domes, JGloptLab–a rigorous global optimization software, in preparation (2017)

    Google Scholar 

  3. F. Domes, M. Fuchs, H. Schichl, A. Neumaier, The optimization test environment. Optim. Eng. 15, 443–468 (2014)

    Article  MathSciNet  Google Scholar 

  4. F. Domes, A. Neumaier, Constraint propagation on quadratic constraints. Constraints 15, 404–429 (2010)

    Article  MathSciNet  Google Scholar 

  5. F. Domes, A. Neumaier, Rigorous filtering using linear relaxations. J. Glob. Optim. 53, 441–473 (2012)

    Article  MathSciNet  Google Scholar 

  6. F. Domes, A. Neumaier, Rigorous verification of feasibility. J. Glob. Optim. 61, 255–278 (2015)

    Article  MathSciNet  Google Scholar 

  7. F. Domes, A. Neumaier, Constraint aggregation in global optimization. Math. Program. 155, 375–401 (2016)

    Article  MathSciNet  Google Scholar 

  8. D.I. Doser, K.D. Crain, M.R. Baker, V. Kreinovich, M.C. Gerstenberger, Estimating uncertainties for geophysical tomography. Reliab. Comput. 4(3), 241–268 (1998)

    Article  Google Scholar 

  9. E. Hyvönen, Constraint reasoning based on interval arithmetic: the tolerance propagation approach. Artif. Intell. 58, 71–112 (1992)

    Article  MathSciNet  Google Scholar 

  10. R.B. Kearfott, M.T. Nakao, A. Neumaier, S.M. Rump, S.P. Shary, P. van Hentenryck, Standardized notation in interval analysis, in Proceedings of the XIII Baikal International School-seminar “Optimization methods and their applications”, vol. 4 (Irkutsk: Institute of Energy Systems, Baikal, 2005), pp. 106–113

    Google Scholar 

  11. V. Kreinovich, Interval computations and interval-related statistical techniques: tools for estimating uncertainty of the results of data processing and indirect measurements. Data Model. Metrol. Test. Meas. Sci. 1–29 (2009)

    Google Scholar 

  12. V. Kreinovich, Decision making under interval uncertainty (and beyond), in Human-Centric Decision-Making Models for Social Sciences, (Springer, Berlin, 2014), pp. 163–193

    Google Scholar 

  13. V. Kreinovich, Solving equations (and systems of equations) under uncertainty: how different practical problems lead to different mathematical and computational formulations. Granul. Comput. 1(3), 171–179 (2016)

    Article  MathSciNet  Google Scholar 

  14. V. Kreinovich, Decision Making Under Interval Uncertainty (to appear) (De Gruyter, Berlin, 2018)

    MATH  Google Scholar 

  15. V. Kreinovich, A. Bernat, Parallel algorithms for interval computations: an introduction. Interval Comput. 3, 3–6 (1994)

    MATH  Google Scholar 

  16. V. Kreinovich, P. Patangay, L. Longpré, S.A. Starks, C. Campos, S. Ferson, L. Ginzburg, Outlier detection under interval and fuzzy uncertainty: algorithmic solvability and computational complexity, in Fuzzy Information Processing Society, 2003. NAFIPS 2003. 22nd International Conference of the North American, (IEEE, 2003), pp. 401–406

    Google Scholar 

  17. V. Kreinovich, S.P. Shary, Interval methods for data fitting under uncertainty: a probabilistic treatment. Reliab. Comput. 23, 105–140 (2016)

    MathSciNet  Google Scholar 

  18. T. Montanher, F. Domes, H. Schichl, A. Neumaier, Using interval unions to solve linear systems of equations with uncertainties. BIT Numer. Math. 1–26 (2017)

    Google Scholar 

  19. R.E. Moore, Interval Arithmetic and Automatic Error Analysis in Digital Computing. Ph.D. thesis, (Stanford University, Stanford, CA, USA, 1963)

    Google Scholar 

  20. A. Neumaier, Interval Methods for Systems of Equations, vol. 37, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1990)

    Google Scholar 

  21. A. Neumaier, Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 1004, 271–369 (2004)

    Article  MathSciNet  Google Scholar 

  22. H. Schichl, F. Domes, T. Montanher, K. Kofler, Interval unions. BIT Numer. Math. 1–26 (2016)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. O. Shcherbina, A. Neumaier, D. Sam-Haroud, X.-H. Vu, T.V. Nguyen, Benchmarking global optimization and constraint satisfaction codes, in Global Optimization and Constraint Satisfaction, ed. by C. Bliek, C. Jermann, A. Neumaier, (Springer, Berlin, 2003) pp. 211–222

    Google Scholar 

  25. V. Telerman, D. Ushakov, Data types in subdefinite models, in International Conference on Artificial Intelligence and Symbolic Mathematical Computing, (Springer, Berlin, 1996), pp. 305–319

    Google Scholar 

  26. I.D. Walker, C. Carreras, R. McDonnell, G. Grimes, Extension versus bending for continuum robots. Int. J. Adv. Robot. Syst. 3(2), 171–178 (2006)

    Article  Google Scholar 

  27. A.G. Yakovlev, Computer arithmetic of multiintervals. Probl. Cybern. 66–81 (1987)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank Dr. Ali Baharev for his valuable comments on the manuscript.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Ferenc Domes, Tiago Montanher, Hermann Schichl & Arnold Neumaier

Authors
  1. Ferenc Domes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tiago Montanher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hermann Schichl
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Arnold Neumaier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tiago Montanher .

Editor information

Editors and Affiliations

  1. Department of Teacher Education, University of Texas at El Paso, El Paso, TX, USA

    Olga Kosheleva

  2. Institute of Computational Technologies SB RAS, Novosibirsk, Russia

    Sergey P. Shary

  3. Johns Creek, GA, USA

    Gang Xiang

  4. Deptartment of Informatics, The State Russian Museum, Saint Petersburg, Russia

    Roman Zapatrin

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Domes, F., Montanher, T., Schichl, H., Neumaier, A. (2020). Rigorous Global Filtering Methods with Interval Unions. In: Kosheleva, O., Shary, S., Xiang, G., Zapatrin, R. (eds) Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications. Studies in Computational Intelligence, vol 835. Springer, Cham. https://doi.org/10.1007/978-3-030-31041-7_14

Download citation

Publish with us

Policies and ethics

Search

Navigation