Decomposition Methods for Integer Programming

This article reviews both traditional and integrated decomposition methods for solving mixed-integer linear programs. These methods attempt to exploit tractable substructures of the problem in order to obtain improved solution procedures. The goal is to derive improved methods of bounding the optimal solution value, which can then be used to drive a branch-and-bound algorithm. Such methods are the preferred solution approaches for a wide range of important models. To expose the desired substructure, a common approach is to relax a set of complicating constraints. This is the approach taken by the Dantzig–Wolfe decomposition, Lagrangian relaxation, and cutting-plane methods. Substructure can also be exposed by relaxing the values of a set of variables, that is, considering restrictions of the original problem. This is the approach taken by Benders' decomposition. This article reviews decomposition methodologies based on relaxation of constraints and examines how they are used to solve mixed-integer linear programs. Keywords: Lagrangian relaxation; cutting-planes; column generation; Dantzig–Wolfe decomposition; branch-and-price; relax-and-cut