Revisiting Lagrangian relaxation for network design

Abstract

We consider a general network design model for which we compare theoretically different Lagrangian relaxations. Fairly general assumptions on the model are proposed, allowing us to generalize results obtained for special cases. The concepts are illustrated on two classical network design problems, the fixed-charge multicommodity capacitated network design problem and the single-source capacitated facility location problem. We also show that, even though our assumptions might not be verified, it is often possible to derive extended reformulations that satisfy them. We study the impact of such extended reformulations on Lagrangian relaxations, by focusing on two particular cases, the hop-constrained fixed-charge multicommodity capacitated network design problem and the multicommodity capacitated network design problem.