, Volume 5, Issue 1, pp 75–88 | Cite as

Partial Lagrangian relaxation for general quadratic programming

  • Alain Faye
  • Frédéric RoupinEmail author
Regular Paper


We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.


Quadratic programming Lagrangian relaxations Semidefinite programming 

MSC classification

90C20 90C22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams WP, Sherali HD (1986) A tight linearization and an algorithm for zero-one quadratic progamming problems. Manage Sci 32(10):1274–1290CrossRefGoogle Scholar
  2. Billionnet A (2005). Different formulations for solving the heaviest k-subgraph problem. Inf Syst Oper Res, 43(3):171–186Google Scholar
  3. Helmberg C, Rendl F, Weismantel R (2000) A semidefinite approach to the Quadratic Knapsack Problem. J Comb Optim 4:197–215CrossRefGoogle Scholar
  4. Laurent M (2003) A comparison of the Sherali-Adams, Lovasz-Schrijver, and Lasserre relaxations for 0–1 programming. Math Oper Res 28:470–496CrossRefGoogle Scholar
  5. Lemaréchal C (2003), The omnipresence of Lagrange. 4’OR 1:7–25Google Scholar
  6. Lemaréchal C, Oustry F (2001) Semidefinite relaxations in combinatorial optimization from a lagrangian point of view. In: Hadjisavvas N, Pardalos PM, (eds), Advances in convex analysis and global optim. Kluwer, Dordrecht, pp. 119–134Google Scholar
  7. Lovász L, Schrijver A (1991) Cones of matrices and set-functions and 0–1 optimization. SIAM J Optim 1:166–190CrossRefGoogle Scholar
  8. Luenberger DG (1989) Linear and nonlinear programming. Addison Wesley, ReadingGoogle Scholar
  9. Peressini AL, Sullivan FE, Uhl Jr. JJ (1988) The mathematics of nonlinear programming. Undergraduate Texts in Mathematics, Springer, Berlin Heidelberg New yorkGoogle Scholar
  10. Poljak S, Rendl F, Wolkowicz H (1995) A recipe for semidefinite relaxations for (0,1)-quadratic programming. J Global Optim 7:51–73CrossRefGoogle Scholar
  11. Roupin F (2004) From linear to semidefinite programming: an algorithm to obtain semidefinite relaxations for bivalent quadratic problems. J Comb Optim 8(4): 469–493CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.CEDRICCNAM-IIEEvry CedexFrance

Personalised recommendations