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Relationship among Benders, Dantzig–Wolfe, and Lagrangian Optimization

  1. Churlzu Lim

Published Online: 15 FEB 2011

DOI: 10.1002/9780470400531.eorms0717

Wiley Encyclopedia of Operations Research and Management Science

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. .

Author Information

  1. University of North Carolina at Charlotte, Charlotte, North Carolina

Publication History

  1. Published Online: 15 FEB 2011

Abstract

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.

Keywords:

  • linear programming;
  • Dantzig–Wolfe decomposition;
  • Benders decomposition;
  • Lagrangian optimization;
  • column generation;
  • cutting plane algorithm