Reformulations and Relaxations

To take advantage of the special structure occurring in a formulation of an integer program, it may be desirable to use a decomposition approach. For example, when the constraints can be partitioned into a set of nice constraints and the remaining set of complicating constraints, a Lagrangian approach may be appropriate. The Lagrangian dual provides a bound that can be stronger than that obtained by solving the usual linear programming relaxation; such a bound may be attractive in a branch-and-bound algorithm. An alternative to the Lagrangian approach is a Dantzig–Wolfe reformulation; when used within the context of an enumeration algorithm, this approach is known as branch-and-price. When it is the variables that can be partitioned into nice and complicating variables, a Benders reformulation may be appropriate.