Ann Oper Res (2007) 153: 9–27
The omnipresence of Lagrange
Published online: 17 May 2007
© Springer Science+Business Media, LLC 2007
Abstract Lagrangian relaxation is usually considered in the combinatorial optimization
community as a mere technique, sometimes useful to compute bounds. It is actually a very
general method, inevitable as soon as one bounds optimal values, relaxes constraints, con-
vexiﬁes sets, generates columns, etc. In this paper we review this method, from both points
of view of theory (to dualize a given problem) and algorithms (to solve the dual by non-
Keywords Combinatorial optimization · Lagrange relaxation · Duality · Column
This paper is devoted to Lagrangian relaxation. Its earlier version (Lemaréchal 2003)was
written in the spirit of (Lemaréchal 2001), which was itself inspired from (Hiriart-Urruty
and Lemaréchal 1993); Chap. XII of this latter work is devoted to the theory of Lagrangian
relaxation, and its Chap. XV gives a detailed account of bundle methods. For a simpliﬁed
account in the framework of combinatorial optimization, we can suggest (Geoffrion 1974;
Reeves 1993, Chap. 6) among others.
1 The basic idea
Consider an optimization problem, which we write abstractly as:
sup f(x), x∈ X, c(x) = 0∈ R
(x)= 0,j= 1,...,m]. (1)
We will call it the primal problem.
This is an updated version of the paper that appeared in 4OR, 1(1), 7–25 (2003).
C. Lemaréchal (
655 avenue de l’Europe, Montbonnot, 38s334 Saint Ismier, France