Advertisement

Optimization Letters

, Volume 1, Issue 3, pp 245–258 | Cite as

On a decomposition method for nonconvex global optimization

  • Hoang TuyEmail author
Original Paper

Abstract

A rigorous foundation is presented for the decomposition method in nonconvex global optimization, including parametric optimization, partly convex, partly monotonic, and monotonic/linear optimization. Incidentally, some errors in the recent literature on this subject are pointed out and fixed.

Keywords

Nonconvex global optimization Parametric optimization Partly convex Partly monotonic Monotonic/linear problems Branch and bound decomposition algorithm Lagrangian bound Duality gap Convergence conditions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ben-Tal A., et al. (1994) Global minimization by reducing the duality gap. Math. Program. 63, 193–212CrossRefMathSciNetGoogle Scholar
  2. 2.
    Dür M. (2001) Dual bounding procedures lead to convergent branch and bound algorithms. Math. Program. Ser. A 91, 117–125zbMATHGoogle Scholar
  3. 3.
    Dür M., Horst R. (2002) Lagrange duality and partitioning techniques in nonconvex global optimization. J. Optim. Theory Appl. 95, 347–369CrossRefGoogle Scholar
  4. 4.
    Ekeland I., Temam R. (1976) Convex analysis and variational problems. North-Holland, Amsterdam, American Elsevier, New YorkGoogle Scholar
  5. 5.
    Horst R., Thoai N.V. (2005) Duality bound methods in global optimization. In: Audet C., Hansen P., Savard G. (eds) Essays and Surveys in Global Optimization. Springer, Berlin Heidelberg New York, pp. 79–105CrossRefGoogle Scholar
  6. 6.
    Thoai N.V. (2000) Duality bound method for the general quadratic programming problem with quadratic constraints. J. Optim. Theory Appl. 107, 331–354zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Thoai N.V. (2002) Convergence of duality bound method in partly convex programming. J. Global Optim. 22, 263–270zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Thoai N.V. (2002) Convergence and application of a decomposition method using duality bounds for nonconvex global optimization. J. Optim. Theory Appl. 113, 165–193zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Tuy H. (2004) Minimax theorems revisited. Acta Math. Vietnamica 29, 217–229zbMATHMathSciNetGoogle Scholar
  10. 10.
    Tuy, H.: Convex Analysis and Global Optimization. Kluwer (1998)Google Scholar
  11. 11.
    Tuy H. (2005) On solving nonconvex global optimization by reducing the duality gap. J. Global Optim. 32, 349–365zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Tuy, H.: A parametric minimax theorem with applications. SubmittedGoogle Scholar
  13. 13.
    Tuy H. (2000) Monotonic optimization: problems and solution methods. SIAM J. Optim. 11(2): 464–494zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Tuy H., Al-Khayyal F., Thach P.T. (2005) Monotonic optimization: branch and cut methods. In: Audet C., Hansen P., Savard G. (eds) Essays and Surveys in Global Optimization. Springer, Berlin Heidelberg New York, pp. 39–78CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam

Personalised recommendations