Decomposition and Dynamic Cut Generation in Integer Linear Programming

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.