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On refinement of the unit simplex using regular simplices

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Abstract

A natural way to define branching in branch and bound (B&B) for blending problems is bisection. The consequence of using bisection is that partition sets are in general irregular. The question is how to use regular simplices in the refinement of the unit simplex. A regular simplex with fixed orientation can be represented by its center and size, facilitating storage of the search tree from a computational perspective. The problem is that a simplex defined in a space with dimension \(n>3\) cannot be subdivided into regular subsimplices without overlapping. We study the characteristics of the refinement by regular simplices. The main challenge is to find a refinement with a good convergence ratio which allows discarding simplices in an overlapped and already evaluated region. As the efficiency of the division rule in B&B algorithms is instance dependent, we focus on the worst case behaviour, i.e. none of the branches are pruned. This paper shows that for this case surprisingly an overlapping regular refinement may generate less simplices to be evaluated than longest edge bisection. On the other hand, the number of evaluated vertices may be larger.

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References

  1. Aparicio, G., Casado, L.G., Tóth, B.G., Hendrix, E.M.T., García, I.: On the minimim number of simplex shapes in longest edge bisection refinement of a regular \(n\)-simplex. Informatica 26(1), 17–32 (2015)

    Article  MathSciNet  Google Scholar 

  2. Casado, L.G., García, I., Tóth, B., Hendrix, E.M.T.: On determining the cover of a simplex by spheres centered at its vertices. J. Glob. Optim. 50(4), 645–655 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Casado, L.G., Hendrix, E.M.T., García, I.: Infeasibility spheres for finding robust solutions of blending problems with quadratic constraints. J. Glob. Optim. 39(2), 215–236 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Claussen, J., Žilinskas, A.: Subdivision, sampling and initialization strategies for simplical branch and bound in global optimization. Comput. Math. Appl. 44, 943–955 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. de Klerk, E.: The complexity of optimizing over a simplex, hypercube or sphere: a short survey. CEJOR 16(2), 111–125 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dickinson, P.J.C.: On the exhaustivity of simplicial partitioning. J. Glob. Optim. 58(1), 189–203 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hannukainen, A., Korotov, S., Křížek, M.: On numerical regularity of the face-to-face longest-edge bisection algorithm for tetrahedral partitions. Science of Computer Programming, 90, Part A(0):34 – 41, (2014). Special issue on Numerical Software: design, analysis and verification

  8. Hendrix, E.M.T., Casado, L.G., Amaral, P.: Global optimization simplex bisection revisited based on considerations by Reiner Horst. In: Computational Science and Its Applications ICCSA 2012, volume 7335 of Lecture Notes in Computer Science, pp. 159–173. Springer, Berlin, (2012)

  9. Hendrix, E.M.T., Casado, L.G., García, I.: The semi-continuous quadratic mixture design problem: description and branch-and-bound approach. Eur. J. Oper. Res. 191(3), 803–815 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Horst, R.: On generalized bisection of \(n\)-simplices. Math. Comput. 66(218), 691–698 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Horst, R.: Bisecton by global optimization revisited. J. Optim. Theory Appl. 144(3), 501–510 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Korotov, S., Kek, M.: Red refinements of simplices into congruent subsimplices. Comput. Math. Appl. 67(12), 2199–2204 (2014)

    Article  MathSciNet  Google Scholar 

  13. Žilinskas, J., Paulavičius, R.: Simplicial Global Optimization. Springer, Berlin (2014)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Differential Equations, Budapest University of Technology and Economics, Budapest, Hungary

    B. G.-Tóth

  2. Operations Research and Logistics, Wageningen University, Wageningen, The Netherlands

    E. M. T. Hendrix

  3. Dept. of Informatics, Agrifood Campus of International Excellence (ceiA3), Universidad de Almería, Almería, Spain

    L. G. Casado

  4. Computer Architecture, Universidad de Málaga, Málaga, Spain

    I. García

Authors
  1. B. G.-Tóth
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  2. E. M. T. Hendrix
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  3. L. G. Casado
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  4. I. García
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Corresponding author

Correspondence to B. G.-Tóth.

Additional information

This work is funded by grants from the Spanish Ministry (TIN2012-37483) and Junta de Andalucía (P11-TIC-7176), in part financed by the European Regional Development Fund (ERDF), Project ICT COST Action TD1207 (EU) and by the Hungarian National Research, Development and Innovation Office – NKFIH, OTKA grant PD115554.

Appendix

Appendix

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Number of evaluated simplices/vertices (left/right); \(n=3\) varying \(\epsilon \)

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Number of evaluated simplices/vertices (left/right); \(n=4\) varying \(\epsilon \)

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Number of evaluated simplices/vertices (left/right); \(n=5\) varying \(\epsilon \)

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Number of evaluated simplices/vertices (left/right); \(n=6\) varying \(\epsilon \)

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Number of evaluated simplices/vertices (left/right); \(n=7\) varying \(\epsilon \)

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G.-Tóth, B., Hendrix, E.M.T., Casado, L.G. et al. On refinement of the unit simplex using regular simplices. J Glob Optim 64, 305–323 (2016). https://doi.org/10.1007/s10898-015-0363-7

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  • DOI: https://doi.org/10.1007/s10898-015-0363-7

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