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Computational Optimization and Applications

, Volume 64, Issue 2, pp 535–555 | Cite as

A generalization of \(\omega \)-subdivision ensuring convergence of the simplicial algorithm

  • Takahito KunoEmail author
  • Tomohiro Ishihama
Article

Abstract

In this paper, we refine the proof of convergence by Kuno–Buckland (J Global Optim 52:371–390, 2012) for the simplicial algorithm with \(\omega \)-subdivision and generalize their \(\omega \)-bisection rule to establish a class of subdivision rules, called \(\omega \)-k-section, which bounds the number of subsimplices generated in a single execution of subdivision by a prescribed number k. We also report some numerical results of comparing the \(\omega \)-k-section rule with the usual \(\omega \)-subdivision rule.

Keywords

Global optimization Strictly convex maximization Branch-and-bound Simplicial algorithm \(\omega \)-subdivision 

Notes

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments, which significantly improved the quality of this article. Takahito Kuno was partially supported by a Grant-in-Aid for Scientific Research (C) 25330022 from the Japan Society for the Promotion of Sciences

References

  1. 1.
  2. 2.
    Horst, R.: An algorithm for nonconvex programming problems. Math. Program. 10, 312–321 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Springer, Berlin (1995)zbMATHGoogle Scholar
  4. 4.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Jaumard, B., Meyer, C.: A simplified convergence proof for the cone partitioning algorithm. J. Global. Optim. 13, 407–416 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jaumard, B., Meyer, C.: On the convergence of cone splitting algorithms with \(\omega \)-subdivisions. J. Optim. Theory Appl. 110, 119–144 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kuno, T., Buckland, P.E.K.: A convergent simplicial algorithm with \(\omega \)-subdivision and \(\omega \)-bisection strategies. J. Global. Optim. 52, 371–390 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kuno, T., Ishihama, T.: A convergent conical algorithm with \(\omega \)-bisection for concave minimization. J. Global. Optim. 61, 203–220 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Locatelli, M.: Finiteness of conical algorithm with \(\omega \)-subdivisions. Math. Program. A 85, 593–616 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Locatelli, M., Raber, U.: On convergence of the simplicial branch-and-bound algorithm based on \(\omega \)-subdivisions. J. Optim. Theory Appl. 107, 69–79 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Locatelli, M., Raber, U.: Finiteness result for the simplicial branch-and-bound algorithm based on \(\omega \)-subdivisions. J. Optim. Theory Appl. 107, 81–88 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications. SIAM, Philadelphia (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  14. 14.
    Thoai, N.V., Tuy, H.: Convergent algorithms for minimizing a concave function. Math. Oper. Res. 5, 556–566 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tuy, H.: Concave programming under linear constraints. Soviet Math. 5, 1437–1440 (1964)zbMATHGoogle Scholar
  16. 16.
    Tuy, H.: Normal conical algorithm for concave minimization over polytopes. Math. Program. 51, 229–245 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tuy, H.: Convex Analysis and Global Optimization. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Graduate School of Systems and Information EngineeringUniversity of TsukubaTsukubaJapan
  2. 2.NS Solutions CorporationTokyoJapan

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