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Interval Tools in Branch-and-Bound Methods for Global Optimization

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Abstract

Interval analysis has been applied with success at solving continuous nonlinear programming problems. In this chapter we review the most relevant research on interval branch-and-bound methods in the period 2011-2021, in particular, bounding rules, discarding/filtering methods, rules for the selection of the next box to be processed and subdivision strategies. No constraint programming techniques nor hybrid proposals are included in the review for the sake of brevity, although we do include a review of the available interval arithmetic libraries for different programming languages as well as software packages with implementations of interval branch-and-bound algorithms. Surprisingly, interval tools have been seldom applied to cope with mixed-integer nonlinear programming problems. This chapter also reviews the proposals in the literature to use interval methods in this type of problems, and suggests how some of the techniques proposed for continuous problems can be adapted for mixed-integer problems.

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Notes

  1. 1.

    C-XSC: http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc.html.

  2. 2.

    PROFIL/BIAS: https://www.tuhh.de/ti3/software/profil.shtml.

  3. 3.

    BOOST: https://www.boost.org/doc/libs/1_77_0/libs/numeric/interval/doc/interval.htm.

  4. 4.

    filib++: https://www2.math.uni-wuppertal.de/org/WRST/software/filib.html.

  5. 5.

    KV: http://verifiedby.me/kv/index-e.html.

  6. 6.

    Moore IA: for the code directly email walter.mascarenhas@gmail.com.

  7. 7.

    MPFI: https://gitlab.inria.fr/mpfi/mpfi.

  8. 8.

    Arbitrary precision arithmetics in C-XSC: http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_software.html#mpfr-mpfi.

  9. 9.

    ACETAF: http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_software.html#acetaf.

  10. 10.

    Slope arithmetic: http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_software.html#slope.

  11. 11.

    ivalDB: http://www.ti3.tuhh.de/, https://sites.google.com/site/pauherrero/IVALDB.zip.

  12. 12.

    Intlab: https://www.tuhh.de/ti3/rump/intlab/.

  13. 13.

    Julia IA: https://github.com/JuliaIntervals/IntervalArithmetic.jl.

  14. 14.

    MPFR IA in Julia: https://github.com/andrioni/Intervals.jl.

  15. 15.

    Taylor models in Julia: https://github.com/JuliaIntervals/TaylorModels.jl.

  16. 16.

    Unums in Julia: https://github.com/JuliaComputing/Unums.jl.

  17. 17.

    PyInterval: https://pypi.org/project/pyinterval/.

  18. 18.

    Toolbox for C-XSC: http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_software.html#cxsc-toolbox.

  19. 19.

    Ibex: http://www.ibex-lib.org/.

  20. 20.

    pyIbex: https://github.com/codac-team/pyIbex.

  21. 21.

    EAGO: https://github.com/PSORLab/EAGO.jl.

  22. 22.

    Coconut: https://www.mat.univie.ac.at/~coconut/coconut-environment/.

  23. 23.

    GloptLab: https://www.mat.univie.ac.at/~dferi/gloptlab.html.

  24. 24.

    TestEnv: http://www.mat.univie.ac.at/~dferi/testenv.html.

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Acknowledgements

This research has been supported by grants from Fundación Séneca (The Agency of Science and Technology of the Region of Murcia, 20817/PI/18) and Junta de Andalucía (P-18-RT-1193), in part financed by the European Regional Development Fund (ERDF).

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Fernández, J., G.-Tóth, B. (2022). Interval Tools in Branch-and-Bound Methods for Global Optimization. In: Salhi, S., Boylan, J. (eds) The Palgrave Handbook of Operations Research . Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-96935-6_8

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