Abstract
Under mild assumptions that are satisfied for many network design models, we show that the Lagrangian dual obtained by relaxing the flow constraints is what we call “quasi-separable”. This property implies that the Dantzig-Wolfe (DW) reformulation of the Lagrangian dual exhibits two sets of convex combination constraints, one in terms of the design variables and the other in terms of the flow variables, the latter being linked to the design variables. We compare the quasi-separable DW reformulation to the standard disaggregated DW reformulation. We illustrate the concepts on a particular case, the budget-constrained multicommodity capacitated unsplittable network design problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Gendron, B.: Revisiting Lagrangian relaxation for network design. Discrete Appl. Math. 261, 203–218 (2019)
Crainic, T.G., Frangioni, A., Gendron, B.: Bundle-based relaxation methods for multicommodity capacitated fixed charge network design. Discrete Appl. Math. 112, 73–99 (2001)
Frangioni, A., Gorgone, E.: Bundle methods for sum-functions with “easy” components: applications to multicommodity network design. Math. Program. A 145, 133–161 (2014)
Holmberg, K., Hellstrand, J.: Solving the uncapacitated network design problem by a Lagrangian heuristic and branch-and-bound. Oper. Res. 46, 247–259 (1998)
Holmberg, K., Yuan, D.: A Lagrangian heuristic based branch-and-bound approach for the capacitated network design problem. Oper. Res. 48, 461–481 (2000)
Kliewer, G., Timajev, L.: Relax-and-cut for capacitated network design. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 47–58. Springer, Heidelberg (2005). https://doi.org/10.1007/11561071_7
Sellmann, M., Kliewe, G., Koberstein, A.: Lagrangian cardinality cuts and variable fixing for capacitated network design. In: Möhring, R., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 845–858. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45749-6_73
Geoffrion, A.M.: Lagrangean relaxation for integer programming. Math. Program. Study 2, 82–114 (1974)
Frangioni, A., Gendron, B.: A stabilized structured Dantzig-Wolfe decomposition method. Math. Program. B 140, 45–76 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Frangioni, A., Gendron, B., Gorgone, E. (2020). Quasi-Separable Dantzig-Wolfe Reformulations for Network Design. In: Baïou, M., Gendron, B., Günlük, O., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2020. Lecture Notes in Computer Science(), vol 12176. Springer, Cham. https://doi.org/10.1007/978-3-030-53262-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-53262-8_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-53261-1
Online ISBN: 978-3-030-53262-8
eBook Packages: Computer ScienceComputer Science (R0)