Relationship among B Enders, D Antzig– W Olfe, and L Agrangian Optimization

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.