Abstract
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.
Similar content being viewed by others
References
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
A. Ben-Tal et al. (1994) ArticleTitleGlobal minimization by reducing the duality gap Mathematical Programming 63 193–212 Occurrence Handle10.1007/BF01582066
M. Dür (2001) ArticleTitleDual bounding procedures lead to convergent branch-and-bound algorithms Mathematical Programming Series A 91 117–20010
J.E. Falk (1969) ArticleTitleLagrange multipliers and nonconvex programs SIAM Journal Control 7 534–545 Occurrence Handle10.1137/0307039
J.E. Falk R.M. Soland (1969) ArticleTitleAn algorithm for separable nonconvex programming problems Management Science 15 550–569
I. Ekeland R. Temam (1976) Convex analysis and variational problems North-Holland, American Elsevier Amsterdam, New York
Floudas, C. (1999), Deterministic Global Optimization, Kluwer.
Shor, N.Z. (1998), Nondifferentiable Optimization and Polynomial Problems, Kluwer.
N.Z. Shor P.I. Stetsenko (1989) Quadratic extremal problems and nondifferentiable optimization Naukova Dumka Kiev
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
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
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
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
H. Tuy (2004) Minimax theorems revisited Institute of Mathematics Hanoi
Tuy, H. (1998), Convex Analysis and Global Optimization, Kluwer.
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.
H. Tuy (2002) Wrong results in D.C. Optimization Institute of Mathematics Hanoi
Author information
Authors and Affiliations
Corresponding author
Rights 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). https://doi.org/10.1007/s10898-004-1947-9
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10898-004-1947-9