Lagrangean decomposition: A model yielding stronger lagrangean bounds

Given a mixed-integer programming problem with two matrix constraints, it is possible to define a Lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems, each having one of the two matrices of the original problem as its constraints. There is one Lagrangean multiplier per variable. We prove that the optimal value of this new Lagrangean dual dominates the optimal value of the Lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement. We show on an example that the resulting bound improvement can be substantial. We show on a complex practical problem how Lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology.