Abstract
Primal and dual decomposition procedures can lead – dependent on the structure of the problem – to efficient procedures for solving mixed—integer programming problems. Benders’ decomposition takes advantage of the primal structure of a problem by temporarily fixing the aggravating integer variables, while dual structures can be exploited by relaxing the complicating constraints in Lagrangian fashion and solving the Lagrangian dual with subgradient optimization, multiplier adjustment methods, or Dantzig-Wolfe decomposition.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beasley, J. E. (1988). An Algorithm for Solving Large Scale Capacitated Warehouse Location Problems. European Journal of Operational Research, 33:314–325.
Cornuejols, G., Sridharan, R. and Thizy, J. M. (1991). A Comparison of Heuristics and Relaxations for the Capacitated Plant Location Problem. European Journal of Operational Research, 50:280–297.
Geoffrion, A. M. and Graves, G. W. (1974). Multicommodity Distribution System Design by Benders Decomposition. Management Science, 20(5):822–844.
Holmberg, K. (1990). On the Convergence of Cross Decomposition. Mathematical Programming, 47:269–296.
Holmberg, K. (1992a). Generalized Cross Decomposition Applied to Nonlinear Integer Programming Problems: Duality Gaps and Convexification in Parts. Optimization,23:341–356.
Holmberg, K. (1992b). Linear Mean Value Cross Decomposition: A Generalization of the Kornai-Liptak Method. European Journal of Operational Research,62:55–73.
Holmberg, K. (1994). Cross Decomposition Applied to Integer Programming Problems: Duality Gaps and Convexification in Parts. Operations Research, 42(4):657–668.
Holmberg, K. (1997). Mean Value Cross Decomposition Applied to Integer Programming Problems. European Journal of Operational Research, 97:124–138.
Holmberg, K. (1999). Mean Value Cross Decomposition Based Branch-and-Bound for Mixed-Integer Integer Programming Problems. In: Kall, P. and Lüthi, H.-J., editors, Operations Research Proceedings 1998, pages 200–209. Springer, Berlin.
Holmberg, K. and Jörnsten, K. O. (1984). Cross Decomposition Applied to the Stochastic Transportation Problem. European Journal of Operational Research, 17:361–368.
Jacobsen, S. K. (1983). Heuristics for the Capacitated Plant Location Model. European Journal of Operational Research, 12:253–261.
Kim, S., Cho, S.-C. and Um, B.-S. (1989). A Simplified Cross Decomposition Algorithm for Multiple Right Hand Choice Linear Programming Journal of the Operational Research Society of Japan, 32(4):441–449.
Kornai, J. and Liptak, T. (1965). Two-Level Planning. Econometrica, 33(1):141–169.
Minoux, M. (1986). Mathematical Programming - Theory and Algorithms. John Wiley & Sons, Chichester et al.
Van Roy, T. J. (1980). Cross Decomposition for Large-Scale,Mixed Integer Linear Programming with Application to Facility Location on Distribution Networks. PhD thesis, Katholieke Universiteit Leuven.
Van Roy, T. J. (1983). Cross Decomposition for Mixed Integer Programming. Mathematical Programming, 25:46–63.
Van Roy, T. J. (1986). A Cross Decomposition Algorithm for Capacitated Facility Location. Operations Research, 34(1):145–163.
Wentges, P. (1994). Standortprobleme mit Berücksichtigung von Kapazitätsrestriktionen: Modellierung und Lösungsverfahren. PhD thesis, University of St. Gallen.
Wentges, P. (1996). Accelerating Benders’ Decomposition for the Capacitated Facility Location Problem. Mathematical Methods of Operations Research, 44(2):267–290.
Wentges, P. (1997). Weighted Dantzig-Wolfe Decomposition for Linear Mixed-Integer Programming. International Transactions in Operational Research, 4(2):151–162.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wentges, P. (2001). On Cross Decomposition for Mixed-Integer Programming. In: Kischka, P., Möhring, R.H., Leopold-Wildburger, U., Radermacher, FJ. (eds) Models, Methods and Decision Support for Management. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-57603-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-57603-4_3
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-642-63306-5
Online ISBN: 978-3-642-57603-4
eBook Packages: Springer Book Archive