On using approximations of the Benders master problem

In this paper we study some approximate ways of solving the Benders master problem in Benders' decomposition. The applications of Lagrange duality and surrogate duality to the Benders master problem are found to be equivalent. We also find that the Lagrangean dual of the Benders master problem cannot yield better bounds than the (partial) Lagrangean dual of the original problem. Compared with solving the Benders master problem exactly, the approximations often yield worse bounds and a lack of controllability, which might prohibit the generation of necessary Benders cuts. For an MILP problem this might yield worse bounds than its LP relaxation. A convergence test originating from cross decomposition can be used to check the progress of a method using such approximations.