Relationship among Benders, Dantzig–Wolfe, and Lagrangian Optimization
Published Online: 15 FEB 2011
Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.
Wiley Encyclopedia of Operations Research and Management Science
How to Cite
Lim, C. 2011. Relationship among Benders, Dantzig–Wolfe, and Lagrangian Optimization. Wiley Encyclopedia of Operations Research and Management Science. .
- Published Online: 15 FEB 2011
In this article, we present the equivalence of Benders, Dantzig–Wolfe, and Lagrangian optimization methods for solving linear programs. In particular, we illustrate that applying Dantzig–Wolfe decomposition for solving a linear program is equivalent to employing Benders decomposition to solve its dual linear program, which is in turn equivalent to implementing a cutting plane method to solve its Lagrangian dual problem. We first demonstrate this equivalence when solving a simply structured linear program having a bounded feasible region, and then extend the results to more general cases such as an unbounded feasible region and block diagonal structure.
- linear programming;
- Dantzig–Wolfe decomposition;
- Benders decomposition;
- Lagrangian optimization;
- column generation;
- cutting plane algorithm