Skip to main content
Log in

On Solving Nonconvex Optimization Problems by Reducing The Duality Gap

Journal of Global Optimization Aims and scope Submit manuscript

Cite this article


Lagrangian bounds, i.e. bounds computed by Lagrangian relaxation, have been used successfully in branch and bound bound methods for solving certain classes of nonconvex optimization problems by reducing the duality gap. We discuss this method for the class of partly linear and partly convex optimization problems and, incidentally, point out incorrect results in the recent literature on this subject.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others


  1. F.A. Al-Khayyal C. Larsen T. Voorhis ParticleVan (1995) ArticleTitleA relaxation method for nonconvex quadratically constrained quadratic programs Journal of Global Optimization 6 215–230 Occurrence Handle10.1007/BF01099462

    Article  Google Scholar 

  2. A. Ben-Tal et al. (1994) ArticleTitleGlobal minimization by reducing the duality gap Mathematical Programming 63 193–212 Occurrence Handle10.1007/BF01582066

    Article  Google Scholar 

  3. M. Dür (2001) ArticleTitleDual bounding procedures lead to convergent branch-and-bound algorithms Mathematical Programming Series A 91 117–20010

    Google Scholar 

  4. J.E. Falk (1969) ArticleTitleLagrange multipliers and nonconvex programs SIAM Journal Control 7 534–545 Occurrence Handle10.1137/0307039

    Article  Google Scholar 

  5. J.E. Falk R.M. Soland (1969) ArticleTitleAn algorithm for separable nonconvex programming problems Management Science 15 550–569

    Google Scholar 

  6. I. Ekeland R. Temam (1976) Convex analysis and variational problems North-Holland, American Elsevier Amsterdam, New York

    Google Scholar 

  7. Floudas, C. (1999), Deterministic Global Optimization, Kluwer.

  8. Shor, N.Z. (1998), Nondifferentiable Optimization and Polynomial Problems, Kluwer.

  9. N.Z. Shor P.I. Stetsenko (1989) Quadratic extremal problems and nondifferentiable optimization Naukova Dumka Kiev

    Google Scholar 

  10. N.Z. Shor P.I. Stetsyuk (2002) ArticleTitleLagrangian bounds in multiextremal polynomial and discrete optimization problems Journal of Global Optimization 23 1–41 Occurrence Handle10.1023/A:1014004625997

    Article  Google Scholar 

  11. N.V. Thoai (2000) ArticleTitleDuality bound method for the general quadratic programming problem with quadratic constraints Journal of Optimization Theory and Applications 107 331–354 Occurrence Handle10.1023/A:1026437621223

    Article  Google Scholar 

  12. N.V. Thoai (2002) ArticleTitleConvergence of duality bound method in partly convex programming Journal of Global Optimization 22 263–270 Occurrence Handle10.1023/A:1013871532570

    Article  Google Scholar 

  13. H.D. Tuan P. Apkarian Y. Nakashima (2000) ArticleTitleA new Lagrangian dual global optimization algorithm for solving bilinear matrix inequalities International Journal Robust Nonlinear Control 10 561–578 Occurrence Handle10.1002/1099-1239(200006)10:7<561::AID-RNC493>3.0.CO;2-C

    Article  Google Scholar 

  14. H. Tuy (2004) Minimax theorems revisited Institute of Mathematics Hanoi

    Google Scholar 

  15. Tuy, H. (1998), Convex Analysis and Global Optimization, Kluwer.

  16. Tuy, H. (2000), On Some recent advances and applications of D.C. optimization. In Nguyen, V.H., Strodiot, J.J. and Tossings, P. (eds.) Optimization, Lecture Notes in Economics and Mathematical Systems. Vol. 481, Springer, pp. 473–497.

  17. H. Tuy (2002) Wrong results in D.C. Optimization Institute of Mathematics Hanoi

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Hoang Tuy.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tuy, H. On Solving Nonconvex Optimization Problems by Reducing The Duality Gap. J Glob Optim 32, 349–365 (2005).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: