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.

Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to generate bounds for mixed-integer linear programming problems. Traditionally, these methods have been viewed as distinct from polyhedral methods, in which bounds are obtained by dynamically generating valid inequalities to strengthen a linear programming relaxation. Recently, a number of authors have proposed methods for integrating dynamic cut generation with various decomposition methods to yield further improvement in the computed bounds. In this paper, we describe a framework within which most of these methods can be viewed from a common theoretical perspective. We then show how the framework can be extended to obtain a new technique we call decompose and cut. As a by-product, we describe how these methods can take advantage of the fact that solutions with known structure, such as those to a given relaxation, can frequently be separated much more easily than arbitrary solutions.

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Both cutting plane methods and traditional decomposition methods are procedures that compute a bound on the optimal value of an integer linear program (ILP) by constructing an approximation to the convex hull of feasible solutions. This approximation is obtained by intersecting the polyhedron associated with the continuous relaxation, which has an explicit representation, with an implicitly defined polyhedron having a description of exponential size. In this paper, we first review these classical procedures and then introduce a new class of bounding methods called integrated decomposition methods, in which the bound yielded by traditional approaches is potentially improved by introducing a second implicitly defined polyhedron. We also discuss the concept of structured separation, which is related to the well-known template paradigm for dynamically generating valid inequalities and is central to our algorithmic framework. Finally, we briefly introduce a software framework for implementing the methods discussed in the paper and illustrate the concepts through the presentation of applications.