Journal of Global Optimization

, Volume 47, Issue 3, pp 485–501 | Cite as

\({{\mathcal {D}(\mathcal {C})}}\)-optimization and robust global optimization

  • Hoang TuyEmail author


For solving global optimization problems with nonconvex feasible sets existing methods compute an approximate optimal solution which is not guaranteed to be close, within a given tolerance, to the actual optimal solution, nor even to be feasible. To overcome these limitations, a robust solution approach is proposed that can be applied to a wide class of problems called \({{\mathcal {D}(\mathcal {C})}}\)-optimization problems. DC optimization and monotonic optimization are particular cases of \({{\mathcal {D}(\mathcal {C})}}\)-optimization, so this class includes virtually every nonconvex global optimization problem of interest. The approach is a refinement and extension of an earlier version proposed for dc and monotonic optimization.


Nonconvex global optimization Approximate optimal solution Robust approach Essential optimal solution dc optimization dm (monotonic) optimization \({{\mathcal {D}(\mathcal {C})}}\)-optimization Successive Incumbent Transcending Algorithm 

AMS Subjcet Classfication

90C26 90C30 90C31 90C57 49K40 65K05 


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  1. 1.
    Audet C., Hansen P., Jaumard B., Savard G.: A branch and cut algorithm for nonconvex quadratically constrained quadratic programming. Math. Progr. Ser. A 87, 131–152 (2000)Google Scholar
  2. 2.
    Horst R., Tuy H.: Global Optimization: Deterministic Approaches. 3rd edn. Springer, Berlin (1996)Google Scholar
  3. 3.
    Lasserre J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)CrossRefGoogle Scholar
  4. 4.
    Sherali H., Adams W.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer, Dordrecht (1999)Google Scholar
  5. 5.
    Tikhonov A.N.: On a reciprocity principle. Sov. Math. Dokl. 22, 100–103 (1980)Google Scholar
  6. 6.
    Tuy H.: Convex programs with an additional reverse convex constraint. J. Optim. Theory Appl. 52, 463–486 (1987)CrossRefGoogle Scholar
  7. 7.
    Tuy H.: Convex Analysis and Global Optimization. Kluwer, Dordrecht (1978)Google Scholar
  8. 8.
    Tuy H.: Monotonic optimization: problems and solution approaches. SIAM J. Optim. 11(2), 464–494 (2000)CrossRefGoogle Scholar
  9. 9.
    Tuy H., Hoai Phuong N.T.: A unified monotonic approach to generalized linear fractional programming. J. Glob. Optim. 23, 1–31 (2002)CrossRefGoogle Scholar
  10. 10.
    Tuy H., Thach P.T., Konno H.: Optimization of polynomial fractional functions. J. Glob. Optim. 29, 19–44 (2004)CrossRefGoogle Scholar
  11. 11.
    Tuy, H., Al-Khayyal, F., Thach, P.T.: Monotonic optimization: branch and cut methods. In: Audet, C., Hansen, P., Savard, G. Essays and Surveys on Global Optimization, Springer, Berlin, pp. 39–78 (2005)Google Scholar
  12. 12.
    Tuy H.: Robust solution of nonconvex global optimization problems. J. Glob. Optim. 32, 307–323 (2005)CrossRefGoogle Scholar
  13. 13.
    Tuy H.: Polynomial optimization: a robust approach. Pac. J. Optim. 1, 357–374 (2005)Google Scholar
  14. 14.
    Tuy H., Minoux M., Hoai-Phuong N.T.: Discrete monotonic optimization with application to a discrete location problem. SIAM J. Optim. 17, 78–97 (2006)CrossRefGoogle Scholar
  15. 15.
    Tuy H., Hoai-Phuong N.T.: A robust algorithm for quadratic optimization under quadratic constraints. J. Glob. Optim. 37, 557–569 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam

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