Decomposition and Dynamic Cut Generation in Integer Programming

Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to develop bounds for integer programming problems. We draw connections between these classical approaches and techniques based on generating strong valid inequalities. We also discuss several methods for incorporating dynamic cut generation into traditional decomposition methods and present a new paradigm for separation called decompose and cut. The methods we discuss take advantage of the fact that separation of a solution to a combinatorial relaxation is often much easier than separation of an arbitrary real vector.